Review1 of Liber De Ludo Aleae (Book on Games of Chance) by ...

Review 1 <strong>of</strong> <strong>Liber</strong> <strong>De</strong> <strong>Ludo</strong> <strong>Aleae</strong> (<strong>Book</strong> on Games <strong>of</strong> Chance) by Gerolamo Cardano 

1. Biographical Notes Concerning Cardano 

Gerolamo Cardano (also referred to in the literature as Jerome Cardan), was born in Pavia, in present 

day Italy, in 1501 and died at Rome in 1576. Educated at the universities <strong>of</strong> Pavia and Padua, 

Cardano practised as a medical doctor from 1524 to 1550 in the village <strong>of</strong> Sacco and in Milan. 

During this period he appears to have studied mathematics and other sciences. He published several 

works on medicine and in 1545 published a text on algebra, the Ars Magna. Among his books is the 

<strong>Liber</strong> de <strong>Ludo</strong> <strong>Aleae</strong> (<strong>Book</strong> on Games <strong>of</strong> Chance), written sometime in the mid 1500s, although 

unpublished until 1663. 

2. Review <strong>of</strong> <strong>De</strong> <strong>Ludo</strong> <strong>Aleae</strong> 

Cardano’s text was originally published in Latin, in 1663. An English translation by Sydney Henry 

Gould is provided in Pr<strong>of</strong>essor Oystein Ore’s book Cardano, The Gambling Scholar (Princeton 

University Press, 1953). Pr<strong>of</strong>essor Ore’s book provides both biographical information relating to 

Cardano, as well as commentary on Cardano’s presentation <strong>of</strong> a probability theory relating to dice 

and card games. 

In <strong>De</strong> <strong>Ludo</strong> <strong>Aleae</strong> Cardano provides both advice and a theoretical consideration <strong>of</strong> outcomes in dice 

and card games. The text (as published) is composed <strong>of</strong> 32 short chapters. The present review is 

concerned principally with chapters 9 to 15, illustrating aspects <strong>of</strong> the theory concerning dice. 

The first eight chapters provide a brief commentary on games and gambling, <strong>of</strong>fering advice to 

players, and suggesting both the dangers and benefits in playing. It may be <strong>of</strong> interest to quote some 

<strong>of</strong> Cardano’s comments regarding the playing <strong>of</strong> games <strong>of</strong> chance: 

“...in times <strong>of</strong> great anxiety and grief, it is considered to be not only allowable, but even beneficial.” 

“..in times <strong>of</strong> great fear or sorrow, when even the greatest minds are much disturbed, gambling is far 

more efficacious in counteracting anxiety than a game like chess, since there is the continual 

expectation <strong>of</strong> what fortune will bring.” 

“In my own case, when it seemed to me after a long illness that death was close at hand, I found no little 

solace in playing constantly at dice.” 

“However, there must be moderation in the amount <strong>of</strong> money involved; otherwise, it is certain that no one 

should ever play.” 

“..the losses incurred include lessening <strong>of</strong> reputation, especially if one has formerly enjoyed any 

considerable prestige; to this is added loss <strong>of</strong> time...neglect <strong>of</strong> one’s own business, the danger it may 

become a settled habit, the time spent in planning after the game how one may recuperate, and in 

remembering how badly one has played.” 

1 Submitted for STA 4000H under the direction <strong>of</strong> Pr<strong>of</strong>essor Jeffrey Rosenthal. 

1

Cardano especially warns that lawyers, doctors and those in like pr<strong>of</strong>essions avoid gambling, which could 

be injurious to their reputations and business. Interestingly, he adds: 

“Men <strong>of</strong> these pr<strong>of</strong>essions incur the same judgement if they wish to practice music.” 

In Chapter 6 Cardano presents what he refers to as the Fundamental Principle <strong>of</strong> Gambling: 

“The most fundamental principle <strong>of</strong> all in gambling is simply equal conditions...<strong>of</strong> money, <strong>of</strong> 

situation...and <strong>of</strong> the dice itself. To the extent to which you depart from that equality, if it is in your 

opponent’s favour, you are a fool, and if in your own, you are unjust.” 

What is most important for our purposes, is to recognise that Cardano’s fundamental principle states that 

games <strong>of</strong> chance can only be fairly played when there are equiprobable outcomes. This principle is the 

basis for his theory relating to outcomes in games <strong>of</strong> dice. 

Cardano begins to present (the results) <strong>of</strong> his theory on dice in Chapter 9: On the Cast <strong>of</strong> One Die. Given 

that a die has six points, he states: 

“...in six casts each point should turn up once; but since some will be repeated, it follows that others will 

not turn up.” 

We see here that his principle is at work (the symmetry <strong>of</strong> the die allows equiprobable outcomes), and 

also that he recognises (confirmed by experience no doubt), that the principle is an ideal, and that in 

practice we will not have each point turn up once in every six casts. There would appear to be an implicit 

understanding <strong>of</strong> a “long range relative frequency” interpretation <strong>of</strong> “in six casts each point should turn 

up once”. Or, in the contemporary language <strong>of</strong> probability theory, we would say that we expect in six 

casts each point should turn up once. 

In this chapter, the concepts referred to as “circuit” and “equality” are introduced: 

“One-half <strong>of</strong> the total number <strong>of</strong> faces always represents equality; thus the chances are equal that a given 

point will turn up in three throws, for the total circuit is completed in six, or again that one <strong>of</strong> three given 

points will turn up in one throw. For example, I can as easily throw one, three, or five as two, four, or 

six.” 

The “circuit” refers to the number <strong>of</strong> possible (elementary) outcomes, what in contemporary probability 

theory may be referred to as “the size <strong>of</strong> the sample space”. “Equality” appears to be a concept related to 

expectation. Since a given point on a die is expected to turn up once in six throws (the circuit), it could 

equally turn up in the first or second three casts. Cardano also provides a variation on this interpretation, 

indicating that in one throw, three given points (1,3,5) could turn up as easily as the three other points 

(2,4,6). Equality then can be understood as defined, that is, one-half <strong>of</strong> the circuit, or as (in contemporary 

terms) an event, which is as likely as its complementary event (that is, an event with probability one-half). 

Pr<strong>of</strong>essor Ore suggests that the concept <strong>of</strong> equality is a consequence <strong>of</strong> Cardano having “the practical 

game in mind”: 

“...he seems to assume that usually there are only two [players]...each will stake the same amount A so 

that the whole pot is P = 2A. When a player considers how much he has won or lost it is natural to relate 

it not to the whole pot 2A but to his own stake A. In terms <strong>of</strong> such a measure his expectation becomes 

2

E = pP = 2pA = peA 

[where p refers to the proportion <strong>of</strong> favourable outcomes for one player and pe is called the equality 

proportion by Pr<strong>of</strong>essor Ore] 

so that the equality proportion or the double probability becomes the natural factor measuring loss or 

gain. …In a fair game…the number <strong>of</strong> favourable and unfavourable cases must be the same and each 

player has the same probability [1/2]. …This means that each player has equality in his favourable cases, 

so that the corresponding equality proportions are [1]. And Cardano expresses this simply by saying that 

“there is equality”. 

In Chapter 11 Cardano discusses the case <strong>of</strong> two dice. He enumerates the various possible throws: 

“…there are six throws with like faces, and fifteen combinations with unlike faces, which when doubled 

gives thirty, so that there are thirty-six throws in all, and half <strong>of</strong> these possible results is eighteen.” 

It is somewhat interesting that Cardano does not at this stage provide an illustration, or table to aid his 

explanation. If the outcomes <strong>of</strong> the cast <strong>of</strong> two dice are represented by ordered pairs, the throws with like 

faces are (1,1), (2,2)…(6,6), six in all, and those with unlike faces are: (1,2), (1,3), (1,4), (1,5), (1,6), 

(2,3), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6), (4,5), (4,6), (5,6), fifteen in number, and finally: (2,1)…(6,5), 

another fifteen. In total there are, as Cardano states, 36 possible outcomes. His use <strong>of</strong> the text only 

description surely would have made the subject more difficult for the reader to appreciate his reasoning – 

unless the reader was well versed in the subject matter. A lay reader for example, would likely ask why 

the unlike face combinations would have to be doubled. As it turns out, this manner <strong>of</strong> explanation is 

typical in Cardano’s text, although he does provide some illustrations. For this reason it would seem a 

reasonable conjecture that the work is intended for those persons familiar with gambling. 

In this chapter, a result is given, comparing how likely, relative to equality, it is to get at least one die with 

one point (an ace) in each <strong>of</strong> two casts <strong>of</strong> two dice: 

“The number <strong>of</strong> throws containing at least one ace is eleven out <strong>of</strong> the circuit <strong>of</strong> thirty-six; or somewhat 

more than half <strong>of</strong> equality; and in two casts <strong>of</strong> two dice the number <strong>of</strong> ways <strong>of</strong> getting at least one ace 

twice is more than 1/6 but less than 1/4 <strong>of</strong> equality.” 

Cardano does not describe how he derived this result, however the following reasoning seems quite 

possible. The number <strong>of</strong> ways <strong>of</strong> getting at least one ace is 11 – (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), 

(3,1), (4,1), (5,1) and (6,1). As 11 is less than 12, there are fewer than 12 times 12 or 144 ways <strong>of</strong> getting 

at least an ace in both casts <strong>of</strong> the dice. In two casts <strong>of</strong> two dice, there are 36 times 36 or 1,296 possible 

outcomes (the circuit in this case). Equality is defined as half <strong>of</strong> 1,296, which is 648. 144 divided by 648 

is less than 1/4. Also, since 10 is less than 11, there are more than 10 times 10 or 100 ways <strong>of</strong> getting at 

least an ace in both casts. As 100 divided by 648 is more than 1/6, Cardano’s result is confirmed. Note 

that the statement “more than 1/6 but less than 1/4 <strong>of</strong> equality” in terms <strong>of</strong> modern probability would be 

“more than 1/6 times 1/2 or 1/12 but less than 1/4 times 1/2 or 1/8”, that is the probability <strong>of</strong> the event is 

between 1/8 and 1/12. Why wasn’t Cardano more exact? If the above reasoning was followed, he would 

have known that there were 121 possible ways in which the aces could turn up. Possibly, for his 

purposes, the precise fraction is unnecessary. It may have been sufficient to explain that a game in which 

a player wagers on the occurrence <strong>of</strong> at least one ace in two casts <strong>of</strong> two dice, was not a fair game. The 

player could expect only between 1/4 and 1/6 <strong>of</strong> their own stake. Also, interestingly, the use <strong>of</strong> upper and 

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lower bounds suggests some variation in practice – although we can not say for sure that this is what 

Cardano wished to express. It may have been seen as a more aesthetic way <strong>of</strong> describing the proportion. 

Clearly however, concepts and problems familiar to modern students <strong>of</strong> probability are being considered. 

In Chapter 12, the casting <strong>of</strong> three dice is considered. Again the possible outcomes are enumerated, the 

total (circuit) being 216. While the wording is somewhat vague, Cardano appears to make an error in 

reporting the number <strong>of</strong> outcomes with at least one unspecified point (such as an ace): 

“…out <strong>of</strong> the 216 possible results, each single face will be found in 108 and will not be found in as 

many.” 

According to Pr<strong>of</strong>essor Ore his reasoning seems to be that for one cast <strong>of</strong> a die, 1/6 <strong>of</strong> a given point can 

be expected to turn up. In 3 throws, 3 times 1/6 or 1/2 <strong>of</strong> the point will occur. Of 216 possible outcomes, 

108 would be favourable. However, if Cardano realised (as appears to be the case) that there were 11 

possible outcomes for a single point in two throws <strong>of</strong> the dice, would he make such an error? Was it an 

approximation only? Cardano does indicate in subsequent chapters that there are 91 possible outcomes, 

which is correct. 

Chapters 13 and 14 concern outcomes <strong>of</strong> the sum <strong>of</strong> points. The theory is an extension <strong>of</strong> the earlier 

results, and Cardano observes: 

“In the case <strong>of</strong> two dice, the points 12 and 11 can be obtained respectively as (6,6) and (6,5). The point 

10 consists <strong>of</strong> (5,5) and <strong>of</strong> (6,4) but the latter can occur in two ways, so that the whole number <strong>of</strong> ways <strong>of</strong> 

obtaining 10 will be 1/12 <strong>of</strong> the circuit and 1/6 <strong>of</strong> equality.” 

Note that Cardano uses ordered pairs to illustrate the outcomes (presuming that the translated text does 

not use this for simplification). Also note that the expression 1/12 <strong>of</strong> the circuit is what we would refer to 

as a probability <strong>of</strong> 1/12. 

In Chapter 14 the use <strong>of</strong> the term “odds” is found, as we would apply it today: 

“If therefore, someone should say, ‘I want an ace, a deuce, or a trey, you know that there are 27 

favourable throws, and since the circuit is 36, the rest <strong>of</strong> the throws in which these points will not turn up 

will be 9; the odds will therefore be 3 to 1.’” 

An incorrect computation <strong>of</strong> odds is found in the same chapter: 

“If it is necessary for someone that he should throw at least twice, then you know that the throws 

favourable for it are 91 in number, and the remainder is 125; so we multiplying each <strong>of</strong> these numbers by 

itself and get to 8,281 and 15,625, and the odds are about 2 to 1.” 

Cardano realises that an error has been made, and discusses this in chapter 15: 

“This reasoning seems to be false... for example, the chance <strong>of</strong> getting one <strong>of</strong> any three chosen faces in 

one cast <strong>of</strong> one dice is equal to the chance <strong>of</strong> getting one <strong>of</strong> the other three, but according to this 

reasoning there would be an even chance <strong>of</strong> getting a chosen face each time in two casts, and thus in 

three, and four, which is most absurd. For if a player with two dice can with equal chances throw an even 

and an odd number, it does not follow that he can with equal fortune throw an even number in each <strong>of</strong> 

three successive casts.” 

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It is interesting that after having made this observation, the earlier text is not corrected. Pr<strong>of</strong>essor Ore 

notes that this is typical <strong>of</strong> Cardano’s presentations. The passage is also interesting for its use <strong>of</strong> the words 

“chance” and “fortune” relating to the possible outcomes <strong>of</strong> throws. In the following paragraph the word 

“probability” is used: 

“In comparison where the probability is one half, as <strong>of</strong> even faces with odd, we shall multiply the number 

<strong>of</strong> casts by itself and subtract one from the product, and the proportion which the remainder bears to 

unity will be the proportion <strong>of</strong> the wagers to be staked. Thus, in 2 successive casts we shall multiply 2 by 

itself, which will be 4; we shall subtract 1; the remainder is 3; therefore the player will rightly wager 3 

against 1...” 

Cardano continues to discuss the computation <strong>of</strong> odds in the chapter, however what is <strong>of</strong> principal interest 

is the use <strong>of</strong> the word “probability”. The above passage is quite possibly the first application <strong>of</strong> the word 

in written form, with the meaning comparable to its use in the modern theory (based on symmetry or a 

long-range relative frequency definition). 

It is well known that the theory <strong>of</strong> probability has its origins in questions on gambling. Why is this the 

case? Although people were aware <strong>of</strong> the variable and unpredictable character <strong>of</strong> every day phenomena 

(such as the weather, commodity prices, etc.), games <strong>of</strong> chance lend themselves to a mathematical 

discussion because the universe <strong>of</strong> possibilities is (relatively) easily known and computed, at least for 

simple games. 

Cardano’s text would appear to be the first known mathematical work on the theory <strong>of</strong> probability, 

although published after the more famous correspondence between Pascal and Fermat. 

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Review 1 <strong>of</strong> Sopra Le Scoperte dei Dadi (Concerning an Investigation on Dice) 

1. Biographical Notes 

Galileo Galilei was born in Pisa in 1564. His early education was at the Jesuit monastery <strong>of</strong> Vallombrosa, 

and attended the University <strong>of</strong> Pisa (with the original intention <strong>of</strong> studying medicine). He became a 

pr<strong>of</strong>essor <strong>of</strong> mathematics at Pisa in 1589, and at Padua in 1592. He is famous for his interest in 

astronomy and physics, including hydrostatics, and dynamics (through his study <strong>of</strong> properties relating to 

gravitation). Works published in 1632 include support for the Copernican system. Although the 

publication was approved by the papal censor, it did (to some degree) contradict an edict in 1616, 

declaring the proposition that the sun was the centre <strong>of</strong> the solar system to be false. After an inquiry, 

Galileo was placed under house arrest, and died near Florence in 1642. 

2. Review <strong>of</strong> Sopra Le Scoperte dei Dadi 

Galileo’s brief research summary on dice is believed to have been written between 1613 and 1623 2 . It is a 

response to a request for an explanation about an observation concerning the playing <strong>of</strong> three dice. While 

the possible combinations <strong>of</strong> dice sides totalling 9, 10, 11, and 12 are the same, in Galileo’s words: 

“…it is known that long observation has made dice-players consider 10 and 11 to be more advantageous 

than 9 and 12.” 

Galileo notes in the opening paragraph <strong>of</strong> his article: 

“The fact that in a dice-game certain numbers are more advantageous than others has a very obvious 

reason, i.e. that some are more easily and more frequently made than others…” 

Galileo explains the phenomenon by enumerating the possible combinations <strong>of</strong> the three numbers 

composing the sum, and presents a tabular summary. The principles allowing the enumeration are 

explained: 

“…we have so far declared these three fundamental points; first, that the triples, that is the sum <strong>of</strong> three-dice 

throws, which are made up <strong>of</strong> three equal numbers, can only be produced in one way; second, that the triples which 

are made up <strong>of</strong> two equal numbers and the third different, are produced in three ways; third, that those triples which 

are made up <strong>of</strong> three different numbers are produced in six ways. From these fundamental points we can easily 

deduce in how many ways, or rather in how many different throws, all the numbers <strong>of</strong> the three dice may be formed, 

which will easily be understood from the following table:” 

10 9 8 7 6 5 4 3 

631 6 621 6 611 3 511 3 411 3 311 3 211 3 111 1 

622 3 531 6 521 6 421 6 321 6 221 3 

541 6 522 3 431 6 331 3 222 1 

532 6 441 3 422 3 322 3 

442 3 432 6 332 3 

433 3 333 1 

27 25 21 15 10 6 3 1 


2 Refer to F.N. David’s Games, Gods and Gambling – A History <strong>of</strong> Probability and Statistical Ideas, Dover 

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The top row <strong>of</strong> the table presents the sum <strong>of</strong> the three dice. Galileo does not provide the enumeration for 

sums 11 to 18, indicating earlier in his article that an investigation from 3 to 10 is sufficient because: 

“…what pertains to one <strong>of</strong> these numbers, will also pertain to that which is the one immediately greater.” 

While the wording is awkward, he is referring to the symmetrical nature <strong>of</strong> the problem, however does 

not provide any more explanation. 

The possible triples are shown under the sums, and to the right <strong>of</strong> each is the number <strong>of</strong> combinations for 

the triple. The last row sums those combinations. 

From Galileo’s table, it can be seen that 10 will show up in 27 ways out <strong>of</strong> all possible throws (which 

Galileo does indicate as 216). Since 9 can be found in 25 ways, this explains why it is at a 

“disadvantage” to 10 (even though each sum can be made from 6 different triples). 

The article is <strong>of</strong> interest for its antiquity in the development <strong>of</strong> ideas relating to the science <strong>of</strong> probability. 

Although words like “chance” and “probability” are not directly used, the idea is conveyed by the 

application <strong>of</strong> terms such as “advantage” or “disadvantage”. Combinatorial mathematics, and an 

appreciation for the equipossibility <strong>of</strong> individual events (gained either by a recognition <strong>of</strong> the symmetry 

<strong>of</strong> the die, or observation <strong>of</strong> results) form the building material for early probability science. 

Publications 1962, page 62. 

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Review 1 <strong>of</strong> Correspondence between Pierre de Fermat and Blaise Pascal 


Pierre de Fermat was born in 1601 at Beaumont-de-Lomagne, studied law at the University <strong>of</strong> Toulouse, 

and served there as a judge. He appears to have corresponded a great deal with scientists in Paris, as well 

as with others, including Pascal, about mathematical ideas. His interests included the theory <strong>of</strong> numbers, 

and is well known for the proposition that the equaton x n + y n = z n has no solutions in the positive integers 

(>2). He died at Castres in 1665. 

Blaise Pascal was born at Clermont in 1623, and died in Paris in 1662. In addition to his contribution, 

along with Fermat, to the science <strong>of</strong> probability, he is well known for his work in geometry and 

hydrostatics. Pascal wrote the Essai pour les Coniques, and invented (and sold) a mechanical calculating 

machine. He may be most famous for his philosophical and religious writings, and is the author <strong>of</strong> the 

Pensees. 

2. Review <strong>of</strong> the correspondence 

This summary is primarily concerned with ideas presented in the first two letters <strong>of</strong> a collection <strong>of</strong> 

correspondence written over a period from 1654 to 1660. The first letter in this series is from Fermat to 

Pascal, and is undated, although it was likely written in June or July <strong>of</strong> 1654 (based on the dates <strong>of</strong> 

subsequent correspondence from Pascal). It would seem that Pascal had earlier written to Fermat, 

discussing the problem relating to the division <strong>of</strong> stakes in a wager on a game <strong>of</strong> dice, when the game is 

suspended before completion. The question appears to have been: If a player needs to get 1 point (a 

specific side <strong>of</strong> the die) in eight throws <strong>of</strong> the die, and after the first three throws has not obtained the 

required point, how much <strong>of</strong> the wager should be distributed to each player if they agree to discontinue 

play? 

Fermat’s letter suggests that Pascal reasoned 125/1,296 <strong>of</strong> the wager should be given to the player. Fermat 

disagrees with this, proposing that the player should receive 1/6 <strong>of</strong> the wager. Fermat’s argument is surely 

based on the equal possibility <strong>of</strong> outcomes for points 1 to 6, due to the symmetry <strong>of</strong> the die. 

Fermat distinguishes between an assessed value for a throw not taken, with subsequent continuation <strong>of</strong> 

the game, and the agreed completion <strong>of</strong> play before eight throws. His reasoning is as follows: 

“If I try to make a certain score with a single die in eight throws…[and] we agree that I will not make the 

first throw; then, according to my theory, I must take in compensation 1/6 th <strong>of</strong> the total sum…Whilst if we 

agree further that I will not make the second throw, I must, for compensation, get a sixth <strong>of</strong> the remainder 

which comes to 5/36 th <strong>of</strong> the total sum…If, after this, we agree that I will not make the third throw, I must 

have…a sixth <strong>of</strong> the remaining sum which is 25/216 th <strong>of</strong> the total…And if after that we agree that I will 

not make the fourth throw…I must again have a sixth <strong>of</strong> what is left, which is 125/1,296 th <strong>of</strong> the total, and 

I agree with you that this is the value <strong>of</strong> the fourth throw, assuming that one has already settled for the 

previous throws.” 

The argument can be summarised in the following table: 


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In Fermat’s theory, the player always has a chance <strong>of</strong> obtaining the whole wager (a chance at least proportional to 

the ratio <strong>of</strong> the one number needed and the six sides <strong>of</strong> the die – that is, a one in six chance). The table above shows 

the sequence <strong>of</strong> probabilities associated with the occurrence <strong>of</strong> the needed point on the last throw (1/6, 5/6 . 1/6, 

5/6 . 5/6 . 1/6 etc.). These are the probabilities associated with the negative binomial distribution, having the required 

point occur on the last <strong>of</strong> a sequence <strong>of</strong> 1,2,…,8 throws. 

In his response letter, dated July 29, 1654, Pascal agrees with Fermat’s reasoning, and presents solutions to two 

specific cases <strong>of</strong> the problem <strong>of</strong> points: 

1) The case involving a player needing one more point. 

2) The case in which a player has acquired the first point. 

For the first case, Pascal uses a recursive process to illustrate the solution. He provides the example <strong>of</strong> two players 

wagering 32 pistoles (gold coins <strong>of</strong> various denominations) each, and begins by considering a dice game in which 

three points are needed. The players’ numbers have equal chances <strong>of</strong> turning up. The following table illustrates 

Pascal’s argument: (the ordered pair notation (a,b) refers to the “state” <strong>of</strong> the game at some stage, with player A 

having thrown a points, and player B, b points; the pair {c,d} refers to the division <strong>of</strong> the wager). 

State <strong>of</strong> Game 

Throw Proportion <strong>of</strong> Wager 

distributed 

Division if player A's 

number turns up next 

9 

Division if player B's 


Remainder <strong>of</strong> original 

Wager 

1 1/6 5/6 

2 1/6 (5/6) 25/36 

3 1/6 (25/36) 125/216 

4 1/6 (125/216) 625/1,296 

5 1/6 (625/1296) 3125/7,776 

6 1/6 (3125/7776) 15,625/46,656 

7 1/6 (15625/46656) 78,125/279,936 

8 1/6 (78125/279936) 390,625/1,679,616 

Accumulated 

Totals 

1,288,991/1,679,616 390,625/1,679,616 

(0.77) (0.23) 

Division if players 

agree to suspend the 

game 

(2,1) {64,0} {32,32} {48,16} 

(2,0) {64,0} {48,16}* {56,8} 

*this corresponds to 

the state (2,1) shown 

above 

(1,0) {56,8}** {32,32} {44,20} 

**this corresponds to 


above

The values distributed upon suspension <strong>of</strong> the game, conform to the expected values. 

The argument is an (early) example <strong>of</strong> the application <strong>of</strong> a “minimax” principle. Both players wish to 

maximize the amount they would receive, and minimize their losses. The “motivation” is illustrated by 

the following “pay<strong>of</strong>f matrix”, with the expected proceeds for player A indicated under the relevant 

circumstances. 

Player A 

Rolls a favourable 

number 

Does not roll a 

favourable number 

Player A would like to maximize the row minimums, while player B wishes to minimize the column 

maximums. Both would settle on 48 (for player A). 

After discussing his theory relating to the equitable distribution <strong>of</strong> the wager amount, Pascal presents the 

following rule : 

“…the value (by which I mean only the value <strong>of</strong> the opponent’s money) <strong>of</strong> the last game <strong>of</strong> two is double 

that <strong>of</strong> the last game <strong>of</strong> three and four times the last game <strong>of</strong> four and eight times the last game <strong>of</strong> five, 

etc.” 

Using the recursive procedure applied by Pascal in the case <strong>of</strong> a game <strong>of</strong> three, with a game <strong>of</strong> four, we 

would proceed as follows: 

State <strong>of</strong> Game 

Agreement to 

settle the 

wager 

Division if player A's 


10 

Player B 

No agreement 

to settle the 

wager 

Division if player B's 


Row 

Minimum 

48 64 48 

48 32 32 

Column Maximum 48 64 

Division if players 

agree to suspend the 

game 

(3,2) {64,0} {32,32} {48,16} 

(3,1) {64,0} {48,16}* {56,8} 

*this corresponds to 


above 

(3,0) {64,0} {56,8}** 

**this corresponds to 


above 

{60,4}

Using Pascal’s terminology, the value <strong>of</strong> the last game <strong>of</strong> four is four. 

The rule for distributing a wager <strong>of</strong> 2W (each player providing W), when one <strong>of</strong> the players requires one 

more point, is then 2W-W/2 n , where n represents the number <strong>of</strong> points needed for the game (before play 

commences). 

Pascal suggests that the solution to the second class <strong>of</strong> problems is more complicated: 

“…the proportion for the first game is not so easy to find…[it] can be shown, but with a great deal <strong>of</strong> 

trouble, by combinatorial methods…and I have not been able to demonstrate it by this other method 

which I have just explained to you but only by combinations.” 

A rule is presented, without detailing a pro<strong>of</strong>: 

“Let the given number <strong>of</strong> games be, for example, 8. Take the first eight even numbers and the first eight 

odd numbers thus: 

2, 4, 6, 8, 10, 12, 14, 16 

and 1, 3, 5, 7, 9, 11, 13, 15. 

Multiply the even numbers in the following way: the first by the second, the product by the third, the 

product by the fourth etc.; multiply the odd numbers in the same way… 

The last product <strong>of</strong> the even numbers is the denominator and the last product <strong>of</strong> the odd numbers is the 

numerator <strong>of</strong> the fraction which expresses the value <strong>of</strong> the first one <strong>of</strong> eight games…” 

If each player wagers W, then the distribution <strong>of</strong> the wager after the first throw would be 

W + W(1/2 . 3/4 . 5/6 . … . (2n-1)/2n) 

Where n is the number <strong>of</strong> points required (after getting the first point). 

How can this formula be shown to be reasonable, using elementary principles <strong>of</strong> probability? One 

approach is to examine the possible ways for completing various games after one player acquires the first 

point. The tree diagrams in Exhibit 1 (last page) illustrate three cases. 

Player A has one point and needs in case a) one more point; b) two more points and c) three more points. 

Each connecting line between possible states <strong>of</strong> the game represents an event with probability 1/2, the 

probability <strong>of</strong> going from one state to the next. Using the diagrams, the probabilities for player A 

obtaining the required points may be assessed. For example, in a) there is a 1/2 probability <strong>of</strong> going from 

(1,0) to (2,0), and a 1/2 . 1/2 = 1/4 probability <strong>of</strong> going from (1,0) to (1,1) to (2,1). The probability for 

player A getting two points (given one point) is then 1/2+1/4 = 3/4. Similarly, for case b) the probability 

is 11/16 and for case c) 21/32. 

If these probabilities are used to obtain the expected values for player A, we have: 

In case a) 3/4 . 2W = (2/4 + ¼)2W = W +W(1/2) 

In case b) 11/16 . 2W = (8/16 + 3/16)2W = W + 3/8 . W = W + W(1/2 . 3/4) 

In case c) 21/32 . 2W = (16/32 + 5/32)2W = W + 5/16 . W = W + W(1/2 . 3/4 . 5/6) 

11

The above sequence <strong>of</strong> expected values conforms to Pascal’s rule. 

To illustrate, Pascal considers a game in which a player has obtained 1 point and needs 4 more. He notes 

that at most 8 plays would be required to complete the game (either player A throws 4 more points, or 

player B will throw the required 5). He observes that 1/2 <strong>of</strong> the number <strong>of</strong> combinations <strong>of</strong> 4 from 8, 

divided by a sum consisting <strong>of</strong> this same value, plus the combinations <strong>of</strong> 5,6,7 and 8 from 8, gives the 

same proportion as 1/2 . 3/4 . 5/6 . 7/8 = 35/128. 

This is the case, since in general: 

1/2 . 3/4 . 5/6 . … . (2n-1)/(2n) = (2n-1)!/n!(n-1)! . 1/2 2n-1 , with: 

(2n-1)!/n!(n-1)! = 1/2 . (2n!/n!n!), and 

2 2n-1 = 1/2 . (1+1) 2n , and 

1/ 

2 ⋅ ( 1+ 

1) 

2n 

= 1/ 

2 ⋅ 

2n 

∑ 

i= 

0 

⎛2n 

⎞ 

⎜ ⎟ 

⎝i 

⎠ 

In the July 29 th letter, Pascal also provides two tables indicating a division <strong>of</strong> wagers for games <strong>of</strong> dice 

suspended at different stages. The tables are not accompanied with detailed explanations. Pascal also 

relates the observations, and questions <strong>of</strong> Monsieur de Mere, relating to a game <strong>of</strong> dice requiring (at least) 

one six to turn up in 4 throws. The odds given in favour <strong>of</strong> this event are 671 to 625. Again, the 

computations are not provided, however, they correspond to the probability given by: 

∑ 

i= 

i 

⎟ 

4 4 

⎜ ( 1/ 

6) 

( 5 / 6) 

1 ⎝ ⎠ 

⎛ ⎞ i 4−i 

Also, it is noted that there is a “disadvantage” in throwing two sixes in 24 such plays. Using the above 

formula, summing the combinations <strong>of</strong> 1,2,…24 out <strong>of</strong> 24, with associated probabilities 1/36 and 35/36 

(to the appropriate exponents) the probability for the event can be shown to be about 0.49. That Monsieur 

de Mere noticed in practice this “disadvantage” is remarkable (he must have observed, and / or played, 

many such games). 

The remaining correspondence includes an interesting dispute over the interpretation <strong>of</strong> combinations <strong>of</strong> 

events, used as a means for computing equitable settlements for wagers (establishing the proportion <strong>of</strong> 

funds to be distributed). 

However, for our purposes, at this stage, it is sufficient to appreciate that combinatorial methods, and the 

identification <strong>of</strong> equipossible events, are the cornerstones for the emerging theory <strong>of</strong> probability, with 

early applications for binomial expansions. Instead <strong>of</strong> using the terms “chance” or “probability”, our 

correspondents used words such as “favour” or “advantage” and “disadvantage”, which convey the same 

12

meaning, in the context <strong>of</strong> a gambling environment. We also noted the early decision theory motivation, 

and its influence on what we will later call expected value. 

A survey <strong>of</strong> the literature does give the impression that Fermat and Pascal, got the “die rolling” for the 

mathematical development <strong>of</strong> the science <strong>of</strong> probability. 

13

Exhibit1 

Tree graphs <strong>of</strong> possible ways for completing two player games 

<strong>of</strong> dice after one player has acquired the first point 

a) Game requiring two points 

b) Game requiring three points 

c) Game requiring four points 

14

Review 1 <strong>of</strong> Christiaan Huygen’s <strong>De</strong> Ratiociniis in <strong>Ludo</strong> <strong>Aleae</strong> (On Reasoning or Computing in 

Games <strong>of</strong> Chance) 


Christiaan Huygens was born at The Hague, Netherlands, in 1629. He studied mathematics and law at the 

University <strong>of</strong> Leiden, and at the College <strong>of</strong> Orange in Breda. His father was a diplomat, and it would 

have been the normal practice for Huygens to follow in that vocation. However, he was more interested 

in the natural sciences, and with support from his father he was able to conduct studies and research in 

mathematics and physics. He is well known for his work relating to the manufacturing <strong>of</strong> lenses, which 

improved the quality <strong>of</strong> telescopes and microscopes. He discovered Titan, identified the rings <strong>of</strong> Saturn, 

and invented the first pendulum clock. He resided in Paris for some time, and made the acquaintance <strong>of</strong> 

persons familiar with Fermat and Pascal, and with their correspondence relating to “the problem <strong>of</strong> 

points” and similar concepts concerning games <strong>of</strong> chance. It is believed that Huygens died at The Hague, 

in 1695. 

2. Review <strong>of</strong> the <strong>De</strong> Ratiociniis in <strong>Ludo</strong> <strong>Aleae</strong> 

On Reasoning in Games <strong>of</strong> Chance, is cited in the literature as the first published mathematical treatise on 

the subject <strong>of</strong> probability 2 . The work was first printed in 1657, before the earlier correspondence between 

Fermat and Pascal was published, although clearly influenced by the content <strong>of</strong> those letters. The present 

review uses an English translation printed in 1714 by S. Keimer, at Fleetstreet, London. 

The treatise is composed <strong>of</strong> a brief introduction, entitled The Value <strong>of</strong> Chances; the statement <strong>of</strong> a 

fundamental postulate; 14 propositions, a corollary and a set <strong>of</strong> five problems for the reader to consider. 

The development <strong>of</strong> the theory is very systematic. Introducing the subject, Huygen’s writes: 

“Although in games depending entirely upon Fortune, the Success is always uncertain; yet it may be 

exactly determined at the same time how much more probability there is that [one] should lose than win” 

Games <strong>of</strong> chance have outcomes that are (generally) unpredictable. At the same time, Huygens claims 

that it is possible to make meaningful statements, or measurements, relating to those systems. While the 

concept <strong>of</strong> probability, perhaps even the word itself 3 , is observed within our earlier readings, Huygens’ 

association <strong>of</strong> the phenomena (games <strong>of</strong> chance) with a relative measure <strong>of</strong> chance, is comparable to a 

modern treatment <strong>of</strong> the theory by first defining a random system or process, and the concept <strong>of</strong> 

probability. Having defined the system and the measure, Huygen’s states his fundamental principle: 

“As a Foundation to the following Proposition, I shall take Leave to lay down this Self-evident Truth: 

That any one Chance or Expectation to win any thing is worth just such a Sum, as would procure in the 

same Chance and Expectation at a fair Lay [or wager]”. 

The wording is somewhat difficult follow, however he gives an example, from which it is evident that the 


2 See for example, Ian Hacking’s The Emergence <strong>of</strong> Probability (Cambridge University Press, 1975), page 92 or 

William S. Peters’ Counting for Something – Statistical Principles and Personalities (Springer – Verlag, 1987), 

page 39. 

3 Refer to Gerolamo Cardano’s <strong>De</strong> <strong>Ludo</strong> <strong>Aleae</strong>. 

15

“Expectation” or value <strong>of</strong> a wager, is the mean <strong>of</strong> the possible proceeds: 

“If any one should put 3 Shillings in one Hand, without telling me [which hand it is in], and 7 in the 

other, and give me Choice <strong>of</strong> either <strong>of</strong> them; I say, it is the same thing as if he should give me 5 

Shillings.” 

Although the word “expectation” (the Latin “expectatio” was used in the original work) 1 is used to refer 

to the value <strong>of</strong> a wager, its meaning in this example does correspond to its use in modern probability 

theory. 

The first proposition states: 

“If I expect a or b, and have an equal chance <strong>of</strong> gaining either <strong>of</strong> them, my Expectation is worth 

(a+b)/2.” 

The expectation is the fair value for a wager. How can this value be calculated in a game where the prizes 

are received with equal chance? Huygens reasons as follows: Suppose there is a lottery with two players, 

and each player buys a ticket for x, and that it is agreed that the proceeds are a and 2x-a, then each player 

could just as easily receive a or 2x-a. Setting 2x-a = b, it follows that the value <strong>of</strong> the lottery ticket is x 

=(a+b)/2. 

The second proposition extends the first to the case <strong>of</strong> three prizes, a,b and c, such that x = (a+b+c)/3 is 

the value <strong>of</strong> the expectation; then in the same manner to four prizes, and so on. 

Proposition III states: 

“If the number <strong>of</strong> Chances I have to gain a, be p, and the number <strong>of</strong> Chances I have to gain b, be q. 

Supposing the chances equal; my Expectation will then be worth ap+bq / p+q.” 

This proposition generalizes the expectation to lotteries with prizes having different chances <strong>of</strong> being 

distributed. Huygens gives the following example: 

“If I have 3 Expectations <strong>of</strong> 13 and 2 Expectations <strong>of</strong> 8, the value <strong>of</strong> my Expectation would by this rule be 

11.” 

Propositions IV to IX illustrate solutions to “the problem <strong>of</strong> points”, in a manner analogous to the 

reasoning <strong>of</strong> Pascal, although Huygens is looking more generally at the problem, not using any specific 

type <strong>of</strong> game as an example. Beginning with simple cases, Huygens solves more complicated problems, 

suggesting: 

“The best way will be to begin with the most easy Cases <strong>of</strong> the Kind.” 

To appreciate the form <strong>of</strong> Huygens’ arguments, consider Proposition VII: 

“Suppose I want two Games, and my Adversary four. 

Therefore it will either fall out, that by winning the next Game, I shall want but one more, and he four, or 

by losing it I shall want two, and he shall want three. So that by Schol Prop.5. and Prop.6., I shall have 

an equal Chance for 15/16a or 11/16a, which, by Prop.1 is just worth 13/16a.” 

1 See Ian Hacking, page 95. 

16

Propositions 5 and 6 explain the expectations when one player needs 1 game, and the other 4 games, and 

when one player needs 2 and the other 3 games. Then applying Proposition 1, the expectations are 

effectively averaged to provide the relevant value. A corollary is then given: 

“From whence it appears, that he who is to get two Games, before another shall get four, has a better 

Chance than he is to get one, before another gets two Games. For in this last Case, namely <strong>of</strong> 1 to 2 his 

Share by Prop. 4 is but 3/4a, which is less than 13/16a.” 

This corollary compares the probabilities relevant to the two cases, using the expectations. This form <strong>of</strong> 

comparison has been made by earlier writers, using the “odds” approach 1 . 

In Poposition IX, Huygens provides a table showing the relative chances for three players in various game 

states. 

Propositions IX to XIV present solutions for problems relating to games <strong>of</strong> dice. The solutions rely upon 

the earlier propositions, especially Proposition 3. As an example, Proposition X relates to the rolling <strong>of</strong> a 

single die: 

“To find how many Throws one may undertake to throw the Number 6 with a single Die.” 

Huygens reasons that for the simplest case, one throw, there is 1 chance to get a six, receiving the wager 

proceeds a, and 5 chances to receive nothing, so that by Proposition 3, the expectation is (1 . a + 5 . 0) / 

(1+5) = 1/6a. To compute the expectation for 1 six in two throws, it is noted that if the six turns up on the 

first die, the expectation will again be a. If not, then referring to the simplest case, there is an expectation 

<strong>of</strong> 1/6a. Using Proposition 3, there is one way to receive a, and 5 ways to receive the 1/6a (the sides 1 to 

5 on the die): 

(1 . a + 5 . (1/6a)) / (1+5) = 11/36a 

This corresponds to the six appearing on the first throw with probability 1/6, or on the second throw with 

probability (5/6) . (1/6). In Huygens system, expectations for simpler cases are combined using 

Proposition 3, to solve more complex problems. 

Proposition XIV uses two linear equations to derive the ratio <strong>of</strong> expected values for two players in the 

following case: 

“If my self and another play by turns with a pair <strong>of</strong> Dice upon these Terms, That I shall win if I throw the 

Number 7, or he if he throw 6 soonest, and he to have the Advantage <strong>of</strong> first Throw: To find the 

Proportion <strong>of</strong> our Chances.” 

With total proceeds set at a, Huygens uses a-x to represent the expectation <strong>of</strong> the player to throw first, and 

x for the second player. When it is the first player’s turn, the second player’s expectation is x. Huygens 

reasons that when the second player is to throw, the expectation must be higher (it is conditioned on the 

first player not throwing a 6). He refers to this expectation as y. On the first player’s turn, the second 

player’s expectation (using proposition 3) will be (5 . 0 + 31y)/36 = 31/36y (as there are 5 ways for the first 

player to get 6). It follows that 31/36y = x (or x=36/31y). On the second player’s turn the expectation is 

(6a + 30x) / 36 (there are 6 ways to roll 7) which equals y. Then: 

1 Odds are given in the text by Cardano and in the correspondence between Fermat and Pascal. 

17

(6a +30x) / 36 = 36/31x , 

with solution x = 31/61a. The ratio <strong>of</strong> the expected values x to a-x is then 31:30. 

The text concludes with a set <strong>of</strong> five problems for the reader to solve (also the practice in most 

contemporary texts in mathematics). 

18

Review 1 <strong>of</strong> Dr. John Arbuthnott’s An Argument for Divine Providence, taken from the constant 

Regularity observed in the Births <strong>of</strong> both Sexes 


John Arbuthnott was born in 1667 at Kincardineshire, Scotland. The following passage from Annotated 

Readings in the History <strong>of</strong> Statistics, by H.A. David and A.W.F Edwards, provides interesting 

biographical information: 

John Arbuthnott, physician to Queen Anne, friend <strong>of</strong> Jonathan Swift and Isaac 

Newton…was no stranger to probability…In 1692 he had published (anonymously) a 

translation <strong>of</strong> Huygens’ <strong>De</strong> ratiociniis in ludo aleae (1657) as Of the Laws <strong>of</strong> 

Chance, adding “I believe the Calculation <strong>of</strong> the Quantity <strong>of</strong> Probability might be 

improved to a very useful and pleasant Speculation, and applied to a great many 

events which are accidental, besides those <strong>of</strong> Games.” There exists a 1694 

manuscript <strong>of</strong> Arbuthnott’s which foreshadows his 1710 paper [An Argument for 

Divine Providence]. 

Dr. Arbuthnott died at London in 1735. 

2. Review <strong>of</strong> An Argument for Divine Providence 

Dr. Arbuthnott’s paper is the first in our series <strong>of</strong> readings to apply the developing theory <strong>of</strong> probability to 

phenomena other than games <strong>of</strong> chance. Earlier, Pascal did use probabilistic reasoning in his article “The 

Wager” to advocate a life <strong>of</strong> faith, in an “age <strong>of</strong> reason”. Interestingly, Arbuthnott’s subject is related to 

Pascal’s. 

There are two principal arguments made in the article: 

1) It is not by chance that the number <strong>of</strong> male births is about the same as the number <strong>of</strong> female births. 

2) It is not by chance that there are more males born than females, and in a constant proportion. 

To support the first proposition, application is made <strong>of</strong> the binomial expansion relating to a die with two 

sides marked M (male) and F (female). Essentially, (M+F) n is a model for the possible combinations <strong>of</strong> 

male and female children born. Arbuthnott observes that as n increases, the binomial coefficient 

associated with the term having identical numbers <strong>of</strong> M and F, becomes small compared to the sum <strong>of</strong> 

the other terms. 

“It is visible from what has been said, that with a very great number <strong>of</strong> Dice…(supposing M to denote 

Male and F Female) that in the vast number <strong>of</strong> Mortals, there would be but a small part <strong>of</strong> all the 

possible Chances for its happening at any assignable time, an equal Number <strong>of</strong> Males and Females 

should be born.” 

Arbuthnott is aware that in reality there is variation between the number <strong>of</strong> males and females: 

“It is indeed to be confessed that this Equality <strong>of</strong> Males and Females is not Mathematical but Physical, 


19

which alters much <strong>of</strong> the foregoing Calculation; for in this Case [the number <strong>of</strong> male and female terms] 

…will lean to one side or the other.” 

However, he writes: 

“But it is very improbable (if mere Chance governed) that they would never reach as far as the 

Extremities…” 

While it would be possible to have large differences in the numbers <strong>of</strong> males and females (with 

binomially distributed data having probability 1/2), the probability <strong>of</strong> this becomes very small when n is 

large. Contrary to Arbuthnott’s argument, it could be reasoned that chance would account for the 

approximate equality in numbers <strong>of</strong> males and females. 

The second proposition discounts chance as the cause for the larger number <strong>of</strong> male births observed 

annually. The form <strong>of</strong> the argument is interesting, because it is similar to a test <strong>of</strong> significance. 

Arbuthnott states the Problem: 

“A lays against B, that every Year there shall be born more Males than Females: To find A’s Lot, or the 

Value <strong>of</strong> his Expectation.” 

A hypothesis is being made in the form <strong>of</strong> a wager. Arbuthnott notes that the probability that there are 

more males than females born must be less than 1 /2 (assuming that there is an equal chance for a male or 

female birth). For this “test” however, he sets the chance at 1 /2 (which would result in a higher 

probability), and notes that for the number <strong>of</strong> males to be larger than the number <strong>of</strong> females in 82 

consecutive years (for which he has data on christenings), the lot would be 1/2 82 . The lot would be even 

less if the numbers were to be in “constant proportion”. Since the data do not support B (in every year 

from 1629 to 1710, male christenings outnumber female christenings), Arbutnott reasons: 

“From whence it follows, that it is Art, not Chance, that governs.” 

The hypothesis <strong>of</strong> equal probability is rejected, and Arbuthnott attributes the observed proportions to 

Divine Providence. 

The second argument has been referred to as the first published test <strong>of</strong> significance 1 . 

1 Ian Hacking, The Emergence <strong>of</strong> Probability, page 168. 

20

Review 1 <strong>of</strong> Pierre Remond de Montmort’s On the Game <strong>of</strong> Thirteen 


With reference to Isaac Todhunter’s text, A History <strong>of</strong> the Mathematical Theory <strong>of</strong> Probability2, Pierre 

Remond de Montmort devoted himself to religion, philosophy and mathematics. He served in the 

capacity <strong>of</strong> cathedral canon at Notre-Dame in Paris, from which he resigned, in order to marry. In 1708 

he published his treatise on “chances”, Essai d’Analyse sur les Jeux de Hazards. L.E Maistrov, in his 

text, Probability Theory – A Historical Sketch3, provides the following biographical information: 

2. Review <strong>of</strong> the article 

“Pierre Remond de Montmort (1678-1719) was a French mathematician 

as well as a student <strong>of</strong> philosophy and religion. He was in 

correspondence with a number <strong>of</strong> prominent mathematicians (N. 

Bernoulli, J. Bernoulli, Leibniz, etc.) and was a well-established and 

authoritative member <strong>of</strong> the scientific community. In particular, Leibniz 

selected him as his representative at the commission set up by the Royal 

Society to rule on the controversy between Newton and Leibniz 

concerning priority in the discovery <strong>of</strong> differential and integral 

calculus…His basic work on probability theory was the “Essai 

d’Analyse sur les Jeux de Hazard”. It went through two editions, the first 

<strong>of</strong> which was printed in Paris in 1708…the second…appeared in 1713, 

although Todhunter claims [1714]. The first part contains the theory <strong>of</strong> 

combinatorics; the second discusses certain games <strong>of</strong> chance with cards; 

the third deals with games <strong>of</strong> chance with dice, the fourth part contains 

the solution <strong>of</strong> various problems including the five problems proposed by 

Huygens…” 

The present review concerns an article in the second part <strong>of</strong> Montmort’s Essai d’Analyse, entitled “On the 

Game <strong>of</strong> Thirteen”. An English translation <strong>of</strong> that article is available in Annotated Readings in the 

History <strong>of</strong> Statistics by H.A. Davids and A.W.F. Edwards.4 

In the first section <strong>of</strong> the article, Montmort provides a description <strong>of</strong> the play <strong>of</strong> Thirteen: 

“The players first draw a card to determine the banker. Let us suppose 

that this is Peter and that the number <strong>of</strong> players is as desired. From a 

complete pack <strong>of</strong> fifty-two cards, judged adequately shuffled, Peter 

draws one after the other, calling one as he draws the first card, two as 

he draws the second, three as he draws the third, and so on, until the 


2 A History <strong>of</strong> the Mathematical Theory <strong>of</strong> Probability, page 78, by Isaac Todhunter, Chelsea Publishing Co., New 

York (1965 unaltered reprint <strong>of</strong> the First Edition, Cambridge 1865). 

3 Probability Theory – A Historical Sketch, page 76, by Leonid E. Maistrov, Academic Press Inc., New York 1974. 

4 Annotated Readings in the History <strong>of</strong> Statistics , Springer-Verlag, New York 2001. 

21

thirteenth, calling king. Then, if in this entire sequence <strong>of</strong> cards he has 

not drawn any with the rank he has called, he pays what each <strong>of</strong> the 

players has staked and yields to the player on his right. But if in the 

sequence <strong>of</strong> thirteen cards, he happens to draw the card he calls, for 

example, drawing an ace as he calls one, or a two as he calls two, or a 

three as he calls three, and so on, then he takes all the stakes and begins 

again as before, calling one, then two, and so on…” 

Provision is made in the rules for a new deck <strong>of</strong> cards should the dealer use all the cards in the first set. 

The game <strong>of</strong> Thirteen provides an early example <strong>of</strong> a problem relating to coincidences or matches. 

Montmort describes a method for computing the chance or expectation <strong>of</strong> drawing a card matching the 

number called by the banker. Since the time <strong>of</strong> Cardano, questions relating to expectation have been 

solved by enumerating the favourable cases and the total possible events. Montmort observes: 

“Let the cards with which Peter plays be represented by a,b,c, d, etc… it 

must be noted that these letters do not always find their place in a 

manner useful to the banker. For example, a, b, c produces only one to 

the person with the cards although each <strong>of</strong> these three letters is in its 

place. Likewise, b, a, c, d produces only one win for Peter, although <strong>of</strong> 

the letters c and d is in its place. The difficulty <strong>of</strong> this problem is in 

disentangling how many times each letter is in its place useful and how 

many times it is useless.” 

To solve the problem, Montmort first considers a game with only two cards, an ace and a two. There is 

only one way in which the banker can receive the proceeds <strong>of</strong> the wager, an ace has to be the first card. 

Montmort then computes the expectation, essentially an application <strong>of</strong> Huygens' 3rd Proposition in <strong>De</strong> 

Ratiociniis. If the proceeds total A, then the banker's expectation is (1 . A + 1 . 0) / 2 = 1/2 A. 

The next case considered is a game with three cards, represented by the letters a,b,c. Montmort observes 

that <strong>of</strong> the six possible combinations for the letters (representing the possible orders for dealing the cards), 

four are favourable to the banker (and two are not favourable): 

"...there are two with a in first place; there is one with b in second place, 

a not having been in first place; and one where c is in third place, a not 

having been in first place and b not having been in second place." 

It follows that the expectation is (4 . A + 2 . 0) / 6 = 2/3 A. 

Similarly four and five card games are considered, indicating the expectations as 5/8 A and 19/30 A, 

respectively. The expectations for games with one to five cards allows Montmort to suggest a formula for 

computing the banker's expectation generally, in a recursive manner: 

[g(p-1)+d] / p 

where: 

p is the number <strong>of</strong> cards; 

g is the espectation when there are p-1 cards, and 

d is the expectation when there are p-2 cards. 

22

A table is given showing the expectations for games up to 13 cards. The banker's expectation for the 

game <strong>of</strong> Thirteen is presented as 109,339,663/172,972,800 A. 

Montmort observes that the expectations can be expressed as series in the form: 

1 - 1/(1.2) + 1/(1.2.3) - 1/(1.2.3.4) +..., 

with alternating positive and negative terms, consisting <strong>of</strong> numerators 1 and denominators 1…(p-2)(p-1)p, 

where p is the number <strong>of</strong> cards. The rapid convergence <strong>of</strong> the expectations to a value between 5/8 and 

19/30, does not appear to have been noticed. An examination <strong>of</strong> the recursive formula may have 

indicated that as the number <strong>of</strong> cards (p) in the game increases, the value identified as g (expectation) 

stabilizes, since (p-1)/p gets closer to 1, and d/p becomes small. Montmort is principally interested in 

explaining how the expectation is computed in the game <strong>of</strong> Thirteen, and describing the mathematical 

expressions from which the expectations can be computed. 

In addition to explaining how the expectation is derived, Montmort provides a table showing the number 

<strong>of</strong> possible favourable deals from specific cards in a game. For example, in a game with five cards, there 

are 24 ways for an ace to be dealt as the banker calls one, 18 ways for the two to be dealt as two is called 

and so on. The table can be used for games having up to eight cards. Montmort completes the article 

with some commentary relating to patterns in that table. 

Montmort’s article adds to the variety <strong>of</strong> problems considered by probability science since the time <strong>of</strong> 

Cardano. Montmort has clearly used Huygens approach, and the principle stated in Proposition IV <strong>of</strong> <strong>De</strong> 

Ratiociniis: 

“…the best way will be to begin with the most easy Cases <strong>of</strong> the kind.” 

23

Review 1 <strong>of</strong> James Bernoulli’s Theorem…From Artis Conjectandi 


According to W.W.R. Ball in A Short Account <strong>of</strong> the History <strong>of</strong> Mathematics2: 


“Jacob or James Bernoulli was born at Bâle on <strong>De</strong>cember 27, 1654; in 

1687 he was appointed chair <strong>of</strong> mathematics in the university there; and 

occupied it until his death on August 16, 1705…In his Artis Conjectandi, 

published in 1713, he established the fundamental principles <strong>of</strong> the 

calculus <strong>of</strong> probabilities…His higher lectures were mostly on the theory 

<strong>of</strong> series…” 

In Chapter IV <strong>of</strong> Part IV <strong>of</strong> his text on probability, Artis Conjectandi (published in 1713), James 

Bernoulli writes: 

“Something further must be contemplated here which perhaps no one 

has thought <strong>of</strong> about till now. It certainly remains to be inquired 

whether after the number <strong>of</strong> observations has been increased, the 

probability is increased <strong>of</strong> attaining the true ratio between the numbers 

<strong>of</strong> cases in which some event can happen and in which it cannot happen, 

so that the probability finally exceeds any given degree <strong>of</strong> certainty…”3 

This review summarizes Bernoulli’s proposed solution to this inquiry largely in his own words, as 

presented in Chapter V <strong>of</strong> Part IV <strong>of</strong> Artis Conjectandi. 

For the purpose <strong>of</strong> this summary, an abridged English translation <strong>of</strong> a copy available in German4 has been 

prepared. The Latin, German and abridged English versions are attached for reference. It should be 

noted that in the German and English copies, some mathematical notation differs from the original, using 

instead a more contemporary system. 

Bernoulli begins by presenting a set <strong>of</strong> increasingly complex lemmas which will be used to prove his 

proposition: 

Lemma 1 

Given a set <strong>of</strong> natural numbers 


2 Originally published by MacMillan & Co. Ltd., London 1912, pages 366 and 367. 

3 Translation by Bing Sung (1966). Translations from James Bernoulli. <strong>De</strong>partment <strong>of</strong> Statistics, Harvard 

University, Cambridge, Massachusetts. 

4 Electronic Research Archive for Mathematics: (Ostwald's Klassiker d. exact. Wissensch. No. 107 u. 108.) 

Published: (1899). 

24

… 

0, 1, 2, ..., r-1, r, r+1, ..., r+s 

continued such that the last member is a multiple <strong>of</strong> r+s, for example 

nr+ns, the new set is: 

0, 1, 2, ..., nr-n, ..., nr, ..., nr+n, ..., nr+ns. 

With increasing n, the terms between nr and nr+n or nr-n, similarly the 

terms between nr+n, nr-n or nr+ns and 0 increase. No matter how large 

n is, the terms greater than nr+n will not exceed s-1 times the number <strong>of</strong> 

terms between nr and nr+n. The number <strong>of</strong> terms below nr-n will not 

exceed r-1 times the terms between nr-n and nr. 

Lemma 2 

If r+s is raised to an exponent, then the expansion will have one more 

term than the exponent. 

Lemma 3 

In the expansion <strong>of</strong> the binomial r+s with exponent an integral multiple 

<strong>of</strong> r+s=t, for example n(r+s)=nt, then first, there is a term M, the largest 

value <strong>of</strong> the terms, if the number <strong>of</strong> terms before and after M are in the 

proportion s to r ... the closer terms to M on the left or right are larger 

than the more distant terms. Secondly, the ratio <strong>of</strong> M to a term closer to 

it is smaller than the ratio <strong>of</strong> that closer term to one more distant, 

provided the number <strong>of</strong> intermediate terms is the same. 

Lemma 4 

In the expansion <strong>of</strong> a binomial with exponent nt, n can be made so large 

that the ratio <strong>of</strong> the largest term M to other terms Ln and Rn which are n 

terms to the left or right from M, can be made arbitrarily large. 

The pro<strong>of</strong>s for Lemmas 3 and 4 are detailed in L.E. Maistrov’s book Probability Theory – A Historical 

Sketch1. In both cases binomial expansions <strong>of</strong> the terms are divided, and algebraic simplification 

presents the required limiting results. 

Lemma 5 

In the expansion <strong>of</strong> a binomial with exponent nt, n may be selected so 

large that the ratio <strong>of</strong> the sum <strong>of</strong> all terms from the largest term M to the 

terms Ln and Rn to the sum <strong>of</strong> the remaining terms, may be made 

arbitrarily large. 

1 Probability Theory – A Historical Sketch, pages 72 and 73, by Leonid E. Maistrov, Academic Press Inc., New 

York 1974. 

25

Pro<strong>of</strong> <strong>of</strong> Lemma5 

According to Lemma 4, as n becomes infinitely large, M / Ln becomes 

infinite, then the ratios L1 / Ln+1 , L2 / Ln+2, L3 / Ln+3 become all the more 

infinite. So it then follows that: 

that is, the sum <strong>of</strong> the terms between M and Ln is infinitely greater than 

the sum <strong>of</strong> the terms left <strong>of</strong> Ln. Since by Lemma 1 the number <strong>of</strong> terms 

left <strong>of</strong> Ln exceeds the terms between Ln and M by only (s-1) times (that is, 

a finite number <strong>of</strong> times), and then from Lemma 3 the terms become 

smaller more distant from Ln, then (the sum <strong>of</strong>) all the terms between Ln 

and M (even if M is not included) will be infinitely larger than (the sum) 

left <strong>of</strong> Ln. In the same way ... [for the right side] 

The Proposition is then stated as: 

Let the number <strong>of</strong> favourable cases to the number <strong>of</strong> unfavourable cases be exactly or 

nearly r/s, therefore to all the cases as r/r+s = r/t - if r+s = t - this last ratio is between 

r+1/t and r-1/t. We can show, as many observations can be taken that it becomes more 

probable arbitrarily <strong>of</strong>ten (for example, c - times) that the ratio <strong>of</strong> favourable to all 

observations lies in the range with boundaries r+1/t and r-1/t. 

Bernoulli observes that given a probability r/t for a favourable outcome, and a probability s/t for an 

unfavourable outcome, in nt trials (with t = r+s), the number <strong>of</strong> (possible) events with all favourable 

outcomes, all but one favourable outcomes, all but two favourable outcomes, etc. are 

nt 

r , 

nt ⎞ nt−1 

⎟r 

s 

⎛ 

⎜ 

⎝1 

⎠ 

, 

L 

L + L + L + ... 

+ L 

1 

+ L 

nt ⎞ nt−2 

2 

⎟r 

s 

⎛ 

⎜ 

⎝2 

⎠ 

2 

+ L 

, 

3 

+ ... + L 

n+ 

1 n+ 

2 n+ 

3 

2n 

These correspond to the terms in the expansion <strong>of</strong> r+s, for which a number <strong>of</strong> useful properties were 

established in the lemmas. First, the number <strong>of</strong> trials with nr favourable outcomes and ns unfavourable 

outcomes is M. Next, the number <strong>of</strong> trials with at least nr-n and at most nr+n favourable outcomes is the 

sum <strong>of</strong> terms between the two limits Ln and Rn defined in Lemma 4. Bernoulli can then write: 

Since the binomial exponent can be selected so large that the sum <strong>of</strong> 

terms which are between both bounds Ln and Rn is more than c times the 

sum <strong>of</strong> all the remaining terms outside <strong>of</strong> these bounds, (from Lemmas 4 

and 5), it follows then that the number <strong>of</strong> observations can be taken so 

large that the number <strong>of</strong> trials in which the ratio <strong>of</strong> the number <strong>of</strong> 

favourable cases to all cases will not cross over the bounds (nr+n)/nt 

and (nr-n)/nt or (r+1)/t and (r-1)/t, is more than c times the remaining 

cases, that is, that it is more than c times probable that the ratio <strong>of</strong> the 

26 

n 

= ∞

number <strong>of</strong> favourable to all cases does not cross over the bounds (r+1)/t 

and (r-1)/t. 

James Bernoulli’s solution is apparently the first pro<strong>of</strong> <strong>of</strong> the Law <strong>of</strong> Large Numbers, which informally is 

stated as: 

The law which states that the larger a sample, the nearer its mean is to 

that <strong>of</strong> the parent population from which the sample is drawn. 

27

Review 1 <strong>of</strong> Abraham de Moivre’s A Method <strong>of</strong> approximating the Sum <strong>of</strong> Terms <strong>of</strong> the Binomial 

(a+b) n …From The Doctrine <strong>of</strong> Chances 


According to Isaac Todhunter in his text, A History <strong>of</strong> the Mathematical Theory <strong>of</strong> Probability2: 

“Abraham de Moivre was born at Vitri, in Champagne, in 1667. On 

account <strong>of</strong> the revocation <strong>of</strong> the edict <strong>of</strong> Nantes3, in 1685, he took shelter 

in England, where he supported himself by giving instruction in 

mathematics and answers to questions relating to chances and annuities. 

He died at London in 1754…<strong>De</strong> Moivre was elected a Fellow <strong>of</strong> the 

Royal Society in 1697…It is recorded that Newton himself, in the later 

years <strong>of</strong> his life, used to reply to inquirers respecting mathematics in 

these words: ‘Go to Mr. <strong>De</strong> Moivre, he knows these things better than I 

do’…” 

<strong>De</strong> Moivre is well known for the theorem: 

n 

[cos( θ ) + isin( 

θ )] = cos( nθ 

) + isin( 

nθ 

) 


This review relates to a supplementary article entitled A Method <strong>of</strong> approximating the Sum <strong>of</strong> the Terms <strong>of</strong> 

the Binomial (a+b) n expanded into a Series, from whence are deduced some practical Rules to estimate 

the <strong>De</strong>gree <strong>of</strong> Assent which is to be given to Experiments (referred to as the Approximatio), which appears 

in later editions (after 1733) <strong>of</strong> Abraham <strong>De</strong> Moivre’s text on probabilities, The Doctrine <strong>of</strong> Chances, first 

published in 1718. 

<strong>De</strong> Moivre’s mathematical presentation in the Approximatio begins with a discussion relating to 

approximating the ratio <strong>of</strong> the middle term <strong>of</strong> the binomial (1+1) raised to very large n, to the sum <strong>of</strong> all 

terms (2 n ). It is indicated that this approximation was developed several years earlier. As a result <strong>of</strong> 

contributions from James Stirling, it was found that the approximate ratio could be written as 

where c is the circumference <strong>of</strong> a circle with radius equal to 1. The value <strong>of</strong> c is then 2π. 

<strong>De</strong> Moivre next states: 

2 

nc 


2 A History <strong>of</strong> the Mathematical Theory <strong>of</strong> Probability, page 78, by Isaac Todhunter, Chelsea Publishing Co., New 

York (1965 unaltered reprint <strong>of</strong> the First Edition, Cambridge 1865). 

3 The Edict <strong>of</strong> Nantes was a proclamation by King Henry IV <strong>of</strong> France and Navarre, guaranteeing civil and religious 

rights to the Huguenots. 

28

“…the Logarithm <strong>of</strong> the Ratio which the middle term <strong>of</strong> a high Power 

has to any Term distant from it by an Interval denoted l, would be 

denoted by a very near approximation, (supposing m = 1 / 2n) by the 

Quantities 

(m+l-1/2) x log(m+l-1) + (m-l+1/2) x log(m-l+1) 

- 2m x log m + log ((m+l)/m). ” 

<strong>De</strong> Moivre does not provide details on the derivation <strong>of</strong> the above formulae. Anders Hald describes their 

derivation in his book A History <strong>of</strong> Probability and Statistics and Their Applications before 1750 1 . 

He then presents Corollary 1: 

“This being admitted, I conclude, that if m or 1/ 2n be a Quantity 

infinitely great, then the Logarithm <strong>of</strong> the Ratio, which a Term distant 

from the middle by the Interval l, has to the middle Term, is –2ll/n.” 

Again, the derivation is not shown. It follows noting that: 

(m+l-1/2)log(m+l-1) +(m-l+1/2)log(m-l+1) –2mlogm + log((m+l)/m) 

is equivalent to 

(m+l-1/2)log(m+l-1) - (m+l-1/2)logm +(m-l+1/2)log(m-l+1) - (m-l+1/2)logm + log((m+l)/m). 

Then approximating log(m+l-1) by log(m+l) and log(m-l+1) by log(m-l), for large m, and re-writing the 

above terms using the properties <strong>of</strong> logarithms, we have 

(m+l-1/2)log((m+l)/m) + (m-l+1/2)log((m-l)/m) + log((m+l)/m). 

Recalling that log(1+x) = x – x 2 /2 + x 3 /3 - … when -1 < x ≤ 1, then the above can be approximated, for l 

less than the square root <strong>of</strong> n, by 

(m+l-1/2)(l/m – l 2 /2m 2 ) + (m-l+1/2)(-l/m – l 2 /2m 2 ) + l/m - l 2 /2m 2 = 2ll/n 

Then the approximation to the inverse ratio’s logarithm is –2ll/n. 

In Corollary 2, <strong>De</strong> Moivre notes that the number with “hyperbolic logarithm” (natural logarithm) -2ll/n is: 

2 

6 

1− 

2ll 

/ n + 4l 

/ 2nn 

− 8l 

/ 6n 

+ ... 

This is the series for e -2ll/n , which then approximates the ratio <strong>of</strong> a term l terms distant from the middle 

term, to the middle term. If we represent the middle term by T0, and terms 1, 2, …, l places distant by T1, 

T2, … Tl, then to this point <strong>De</strong> Moivre has obtained: 

T 

2 

0 

≈ 

n 

2 

2πn 

1 Published by John Wiley & Sons, 1990, pages 473 to 476. 

3 

29

and 

T l 

− 2ll 

log( ) ≈ 

T n 

0 

Then Tl = T0e -2ll/n , or as <strong>De</strong> Moivre would write: 

Note that 

2 

T0 ≈ ⋅ 2 

2πn 

n 

for a binomial (1+1) to the exponent n very large. If we consider the binomial (1/2+1/2) n , which is 1 n = 1, 

the middle term would be 

n 

n / 2 

⎟ 

⎛ ⎞ 

⎜ 

⎝ ⎠ 

( 1/ 

2) 

n 

Since in the binomial (1+1) n , the middle term is 

⎛n 

⎞ 

⎜ ⎟ 

⎝n 

/ 2⎠ 

Then this is equivalent to T0, and we can write 

2 n −n 

2 

⋅ 2 ⋅ 2 = 

2πn 

2πn 

for the middle term <strong>of</strong> the binomial (1/2+1/2) n . 

T l 

2 

6 

= T ( 1− 

2ll 

/ n + 4l 

/ 2nn 

− 8l 

/ 6n 

0 

Using the previously defined symbols T0, T1,…, Tl, then sums <strong>of</strong> terms between the middle term and one l 

places distant can be obtained as 

T0 + T1 + T2 + … + Tl = T0 (1+exp(-2 . 1 2 /n) + exp(-2 . 2 2 /n)+…+exp(-2 . l 2 /n)) 

30 

3 

+ ...)

This is essentially what <strong>De</strong> Moivre does in Corollary 2. Using the “hyperbolic logarithm” series 

expansions for each <strong>of</strong> the terms, he develops a sum <strong>of</strong> the binomial terms from the middle, to a term l 

places distant for the case <strong>of</strong> a binomial (1/2 + 1 /2) n : 

2 2 

3 

5 

⋅ ( l − 2l 

/ 1⋅ 

3n 

+ 4l 

/ 2 ⋅ 5n 

2πn 

Setting l = s n , with s = 1/2, 

− ...) 

he gets 

2 

⋅ ( 1/ 

2 −1/ 

3⋅ 

4 + 1/ 

2 ⋅5 

⋅8 

− ...) 

2πn 

Which he observes converges very quickly, and after using a few terms obtains the estimate 0.341344 for 

the sum <strong>of</strong> terms from the middle term to a term, which is about 1/2√n terms distant. Having obtained 

this result, <strong>De</strong> Moivre states in Corollary 3: 

“And therefore, if it was possible to take an infinite number <strong>of</strong> 

Experiments, the Probability that an Event which has an equal number <strong>of</strong> 

Chances to happen or fail, shall neither appear more frequently than 

1/2n + 1/2√n times, not more rarely than 1/2n – 1/2√n times, will be 

expressed by the double Sum <strong>of</strong> the number exhibited in the second 

Corollary, that is by 0.682688…” 

We have here, in 1733, a result about what would later be called a “normal distribution”. In his book, The 

Life and Times <strong>of</strong> the Central Limit Theorem1, William Adams states: 

“<strong>De</strong> Moivre did not name√n/2, which is what we would today call 

standard deviation within the context considered, but in Corollary 6 he 

referred to √n as the Modulus by which we are to regulate our 

estimation.” 

<strong>De</strong> Moivre generalizes the results for the binomial (a+b) n acquiring as William Adams indicates in 

modern notation 

T = 

T e 

l 

0 

2 2 

−( 

a+ 

b) 

l / 2abn 

using the symbols Tl and T0 defined earlier. 

<strong>De</strong> Moivre’s intention was to develop a method for approximating binomial sums or probabilities when 

the number <strong>of</strong> trials was very large. He was able to do this with the aid <strong>of</strong> mathematics relating to series 

1 Kaedmon Publishing Company, New York 1974, page 24. 

31

expansions for logarithms and exponentials, as well as approximation methods for factorials. The 

approximation itself is a normal distribution. In addition to being an early occurrence <strong>of</strong> this distribution, 

for practical application, the approximation is an early illustration <strong>of</strong> a central limit theorem. For large n, 

the middle term (average value) is associated with a normal distribution, and this value can be limited by 

a measure related to √n. 

32

Review 1 <strong>of</strong> Friedrich Robert Helmert’s The Calculation <strong>of</strong> the Probable Error from the Squares <strong>of</strong> 

the Adjusted Direct Observations <strong>of</strong> Equal Precision and Fechner’s Formula2 


Friedrich Robert Helmert was born at Freiberg in 1841. He studied engineering sciences at the technical 

university in Dresden. He was as a lecturer and pr<strong>of</strong>essor <strong>of</strong> geodesy at the technical university in 

Aachen, where he wrote "The Mathematical and Physical Theories <strong>of</strong> Higher Geodesy" (2 volumes, 

Leipzig 1884). In 1886 Helmert became the director <strong>of</strong> the Prussian Geodetic Institute and the 

International Earth Measurement Central Office, as well as pr<strong>of</strong>essor at the university in Berlin. He died 

at Potsdam in 1917. 

2. Review <strong>of</strong> The Calculation <strong>of</strong> Probable Error… 

In this 1876 article by F.R. Helmert, there is apparently for the first time 3 a demonstration that, given 

X1,…,Xn independent N(µ,σ 2 ) random variables, then 

n 

∑ 

i= 

1 

( X − X) 

i 

σ 

2 

2 

is distributed as (what would be called) a chi-square distribution with n-1 degrees <strong>of</strong> freedom. The 

present review is concerned primarily with how Helmert effectively shows this, although his purpose was 

to apply the result to the calculation <strong>of</strong> the mean squared error for σ ) , in order to estimate “the probable 

error” 4 . 

To begin, Helmert considers e1,…,en, the “true errors” <strong>of</strong> a set <strong>of</strong> observations X1,…,Xn. The true errors 

are defined as ei = Xi – µ, where µ is the (true) mean for the population from which the Xi are observed. 

The joint probability “volume” (referred to as the “future probability” 5 ) <strong>of</strong> the ei, given that Xi ~ N(µ,σ 2 ) 

is presented as: 

n 

⎡ h ⎤ 2 

−h 

[ee] 

⎢ ⎥ e de1Lden ⎣ π ⎦ 

where h (referred to as “the precision”) is equal to 1/σ√2 and [ee] is the sum <strong>of</strong> squares <strong>of</strong> the ei: 

e 2 1+e 2 2+…+e 2 n. 


2 An abridged version <strong>of</strong> the article is found in Annotated Readings in the History <strong>of</strong> Statistics pages 109 to 113, by 

H.A. David and A.W.F. Edwards (Springer-Verlag, New York 2001). The section relating to Fechner’s formula has 

been deleted. 

3 Refer to Annotated Readings in the History <strong>of</strong> Statistics page 103, by H.A. David and A.W.F. Edwards (Springer- 

Verlag, New York 2001). 

−1 

4 <strong>De</strong>fined as σΦ (0.75), where Φ is the standard normal distribution function. See Annotated Readings, page 103. 

5 The likelihood function from the set <strong>of</strong> observations. 

33

Helmert notes that [ee] is not known, since the parameter µ is not known (assuming that only sampling <strong>of</strong> 

the population is possible or feasible). 

As the true mean can only be estimated by X , then the true errors are estimated by i X-X, written λi by 

Helmert, and referred to as the “deviations” (from the arithmetic mean <strong>of</strong> the sample). Noting that λ1 + λ2 

+ … + λn = 0, then λn = -λ1 – λ2 - … - λn-1, and with ē = X - µ, the true errors are related to the deviations 

as: 

e1 = λ1 + ē, 

e2 = λ2 + ē 

en-1 = λn-1 + ē 

en = -λ1-…- λn-1 + ē 

Helmert identifies the following matrix with the above transformations: 

⎛1000L 01⎞ 

⎜ 

0100 01 

⎟ 

⎜ 

L 

⎟ 

⎜0010L 01⎟ 

⎜ ⎟ 

⎜ O ⎟ 

⎜0000L 11⎟ 

⎜ ⎟ 

−1−1−1 −1 

1⎟ 

⎝ L ⎠ 

which will be referred to as H. In matrix equation form, the transformation may be written: 

⎛ e ⎞ ⎛1000L 01⎞⎛ 

λ ⎞ 

1 1 

⎜ ⎟ 

e 

⎜ ⎟⎜ 

2 0100 01 λ 

⎟ 

⎜ ⎟ ⎜ 

L 

⎟⎜ 

2 ⎟ 

⎜ e ⎟ ⎜ 3 0010L 01⎟⎜ 

λ ⎟ 3 

⎜ ⎟= ⎜ ⎟⎜ 

⎟ 

⎜ M ⎟ ⎜ O ⎟⎜ 

M ⎟ 

⎜ e ⎟ ⎜0000L 11⎟⎜ 

λ ⎟ 

n−1 n−1 

⎜ 

⎟ 

e ⎟ ⎜ ⎟⎜ 

⎟ 

n −1−1−1L−1 1⎟⎜ e ⎟ 

⎝ ⎠ 

⎝ ⎠⎝ ⎠ 

The determinant <strong>of</strong> the matrix H is n. This can be shown using two properties relating to determinants: 

1) If a matrix M is formed by adding a multiple <strong>of</strong> one column to another 

column in H, then the determinant <strong>of</strong> M equals that <strong>of</strong> H. 

2) The determinant <strong>of</strong> a triangular matrix is equal to the product <strong>of</strong> 

diagonal elements. 

New matrices can be formed by consecutively adding –1 times columns 1 to n-1, to the last column, 

resulting in: 

34

The determinant is then 1 . 1 . … . n = n. 

⎛1000L 00⎞ 

⎜ ⎟ 

⎜ 

0100L 00 

⎟ 

⎜0010L 00⎟ 

⎜ ⎟ 

⎜ O ⎟ 

⎜0000L 10⎟ 

⎜ ⎟ 

−1−1−1 −1 

n ⎟ 

⎝ L ⎠ 

Observing that the Jacobian <strong>of</strong> the transformation <strong>of</strong> variables is equivalent to the determinant <strong>of</strong> H, the 

joint probability for the change <strong>of</strong> variables becomes: 

n 

⎡ h ⎤ 2 2 2 

− h [ λλ ] + h ne 

n⎢ ⎥ e dλ Ldλ 

d 

⎣ π ⎦ 

35 

1 1 e n− 

Helmert then notes that integrating the above expression over all possible values <strong>of</strong> ē results in the 

probability <strong>of</strong> the set λ1,…,λn : 

Then 

n−1 

⎡ h ⎤ 

2 

−h 

[ λλ ] 

n⎢ ⎥ e dλ Ldλ 

⎣ π ⎦ 

n−1 

1 n−1 

⎡ h ⎤ 

2 

−h 

[ λλ ] 

n⎢ ⎥ ∫L∫e dλ Ldλ 

⎣ π ⎦ 

is the probability that [λλ] lies between values u and u+du. 

1 n−1 

Next, a transformation is devised for n-1 new variables t , i = 1,…,n-1, such that [tt] is equivalent to the 

sum <strong>of</strong> n-1 true errors. The transformation in matrix form, is given by:

To illustrate this transformation, consider the two variable case, using: 

Then 

t + t = λ + λ + λ + 2λλ 

+ λ 

2 2 2 2 2 2 

1 2 1 2 1 1 2 2 

⎛ 2 2 2 2 ⎞ 

⎛ t1 

⎞ ⎜ 2 

L ⎟⎛ 

λ1 

⎞ 

⎜ ⎟ ⎜ 2 2 2 2 ⎟⎜ 

⎟ 

⎜ ⎟ ⎜ 3 3 1 3 1 3 1 ⎟⎜ 

⎟ 

⎜ t ⎟ 

2 ⎜ 0 ⋅ ⋅ L ⋅ ⎟⎜ 

λ ⎟ 

2 

⎜ ⎟ ⎜ 2 2 3 2 3 2 3 ⎟⎜ 

⎟ 

⎜ ⎟ ⎜ 4 4 1 4 1 ⎟⎜ 

⎟ 

⎜ t ⎟ 

3 = ⎜ 0 0 

⋅ L ⋅ ⎟⎜ 

λ ⎟ 

3 

⎜ ⎟ ⎜ 3 3 4 3 4 ⎟⎜ 

⎟ 

⎜ ⎟ ⎜ ⎟⎜ 

⎟ 

⎜ ⎟ ⎜ ⎟⎜ 

⎟ 

⎜ 

M 

⎟ ⎜ 

M 

⎜ O 

⎟ ⎟ 

⎜ ⎟ ⎜ ⎟⎜ 

⎟ 

⎜ ⎟ n 1 

t ⎜ ⎜ ⎟ 

n−1 0 0 0 0 

⎟ 

⎝ ⎠ ⎜ L 

⋅ ⎟⎝λn−1⎠ 

⎝ n−1n⎠ t 

2 

1 2( λ1 

2 ) 

t 

= + 

= 

3 λ 

2 

2 2 

2 2 2 

1 2 3 

36 

λ 

= λ + λ + λ , since λ3 =−λ1− λ2. 

The determinant <strong>of</strong> the transformation is √n, noting that the associated matrix is upper triangular, with 

product <strong>of</strong> the diagonal terms: 

2 ⋅ 

3 

⋅ 

2 

4 

⋅L⋅ 3 

n 

n −1 

= 2⋅ 1 

⋅ 

2 

3⋅ 1 

⋅L⋅ 3 

n−1⋅ 1 

⋅ 

n −1 

n 

= n 

Then the probability that [tt] is between u and u + du is given by 

⎡ h ⎤ 

⎢ ⎥ 

⎣ π ⎦ 

n−1 

∫ ∫ 

2 

−h 

[ tt] 

2 

L e dt Ldt 

1 n−1 

Helmert then refers to a result he obtained in 1875: The probability that the sum [tt] <strong>of</strong> n-1 true errors 

equals u, is given by: 

Γ 

h 

n−1 

n − 1 

( ) 

2 

n−3 

2 

2 

−hu 

⋅u ⋅e 

du

where again, h is “the precision”. Since [tt] = [λλ], the density applies to the sum <strong>of</strong> squares <strong>of</strong> n 

deviations. The above density is Gamma( n-1 ,h) for variable hu. This can seen, recalling that if a random 

2 

variable v ~ Gamma (α,β), then the probability density function can be written as: 

Substituting hu for v, 

α 

β α−1−βv v e 

Γ() 

α 

n − 1 

for α, and h for β: 

2 

n−1 

2 

= 

h 

n−3 

2 ⋅( hu) ⋅e 

Γ 

n − 1 

( ) 

2 

n−1 n−3 

( + ) 

2 2 

n−3 

h 

2 

= ⋅u ⋅e 

Γ 

n − 1 

( ) 

2 

= 

n− 

2 

h 

n−3 

2 ⋅u ⋅e 

Γ 

n − 1 

( ) 

Then the probability associated with volume d(hu) is: 

2 

n − 1 

( ) 

2 

37 

2 

−hu 

2 

−hu 

2 

−hu 

= 

n−2 

h 

n−3 

2 

2 −hu 

⋅u ⋅e 

d( hu) 

Γ 

= 

n−1 

h 

n−3 

2 

2 −hu 

⋅u ⋅e 

du 

Γ 

n − 1 

( ) 

Since hu ~ Gamma( n-1 ,h) , then u ~ Gamma( n-1 ,h) h 

2 

2 

probability density function for u is: 

u 

Then 2 

σ ~ 

result: 

χ − , so that u ~ 

2 

n 1 

2 

n 1 1 u 

u 

2 2 

2 e 2 

n 1 

1 n 1 

2 ( ) 2 

σ 

− 

− 

⋅ 

− 

− − 

Γ 

( ) 

σ 

2 

2 

χn 

− 1 . Recalling that u = [λλ], and that [λλ] = 

2 

σ 

. For h = 1 σ 2, 

it can be shown that the 

n 

∑ 

i= 

1 

( X − X) 

i 

2 

, we have the

n 

∑ 

i= 

1 

( X − X) 

i 

σ 

2 

Helmert does not assign a name to the distribution <strong>of</strong> the sum <strong>of</strong> squares [λλ]. His objective is to use the 

result to estimate the probable error. 

Also <strong>of</strong> interest in the article is the estimation related to the precision h (and thereby σ) in a maximum 

likelihood manner in section 2, such that: 

38 

2 

2 

2h n 

χ 

1 [ λλ] 

= σ = 

−1 

2 

n−1

Review 1 <strong>of</strong> R.A. Fisher’s Inverse Probability 


Ronald Aylmer Fisher was born in London in 1890. He received scholarships to study mathematics, 

statistical mechanics and quantum theory at Cambridge University, where he also studied evolutionary 

theory and biometrics. After graduation, he worked for an investment company and taught mathematics 

and physics at public schools from 1915 to 1919. From 1919 to 1943, he was associated with the 

Rothamsted (Agricultural) Experimental Station, contributing to experimental design theory and the 

development <strong>of</strong> a Statistics <strong>De</strong>partment. In 1943 he became pr<strong>of</strong>essor <strong>of</strong> genetics at Cambridge, 

remaining there until his retirement in 1957. He died at Adelaide in Australia, in 1962. 

2. Review <strong>of</strong> Inverse Probability 

In this article, published in 1930, R.A. Fisher cautions against the application <strong>of</strong> prior probability 

densities for parameter estimation using inverse probability 2 , when a priori knowledge <strong>of</strong> the distribution 

<strong>of</strong> the parameters is not available (e.g., from known frequency distributions). Fisher indicates in the first 

paragraph, that the subject had been controversial for some time, suggesting that: 

“Bayes, who seems to have first attempted to apply the notion <strong>of</strong> 

probability, not only to effects in relation to their causes but also to 

causes in relation to their effects, invented a theory 3 , and evidently 

doubted its soundness, for he did not publish it during his life.” 

Fisher describes the manner in which a (known) prior density can be used in the calculation <strong>of</strong> 

probabilities: 

“Suppose that we know that a population from which our 

observations were drawn had itself been drawn at random from 

a super-population…that the probability that θ1, θ2, θ3,... shall lie 

in any defined infinitesimal range 1 2 3 ... dθ dθ dθ 

is given by 

dF =Ψ( 

θ , θ , θ ,...) dθ dθ dθ 

..., 

1 2 3 1 2 3 

then the probability <strong>of</strong> successive events (a) drawing from the 

super-population a population with parameters having the 

particular values θ1, θ2, θ 3,... 

and (b) drawing from such a 

population the sample values 1 ,..., x x n , will have a joint 

probability 


2 According to H.A. David and A.W.F. Edwards in Annotated Readings in the History <strong>of</strong> Statistics, Springer-Verlag 

2001, page 189: 

“…“Inverse probability”…would not necessarily have been taken to 

refer exclusively to the Bayesian method (which in the paper Fisher 

calls “inverse probability strictly speaking”) but to the general 

problem <strong>of</strong> arguing “inversely” from sample to parameter…” 

3 In the mid-1700s. 

39

Ψ ( θ , θ , θ ,...) dθ dθ dθ ... × Π{ 

φ( x , θ , θ , θ ,...) dx }. 

1 2 3 1 2 3 p 1 2 3 p 

p= 

1 

If we integrate this over all possible values <strong>of</strong> θ1, θ2, θ3,... and 

divide the original expression by the integral we shall then have 

a perfectly definite value for the probability…that 

θ , θ , θ ,... shall lie in any assigned limits.” 

1 2 3 

n 

It is noted that this is a direct argument, which provides the frequency distribution <strong>of</strong> the population 

parameters θ. Fisher’s caution relates to cases in which the function Ψ is not known, and is then taken to 

be constant. He argues that this assumption is as arbitrary as any other, and will have inconsistent results. 

While an example is not given in the Inverse Probability paper, it is helpful to consider an illustration 

provided elsewhere by Fisher, related by Anders Hald in A History <strong>of</strong> Mathematical Statistics From 1750 

to 1930 1 . 

Consider the posterior probability element: 

Then if the parameter ς is defined by 

such that 

θ θ ∝ θ −θ θ ≤θ ≤ 

a n−a P( |a,n)d (1 ) d , 0 1 

1 1 

sin ς = 2θ-1, - π ≤ς≤ π 

2 2 

ς = arcsin(2θ −1) 

and ς is assumed to be uniformly distributed, the posterior probability element becomes: 

Since 

it follows that: 

ς ς ∝ ς − ς ς 

a 

n−a P( |a,n)d (1+sin ) (1 sin ) d 

dς darcsin(2θ 

−1) 

= 

dθ dθ 

1 d(2θ−1) 

= ⋅ 

2 

1 −(2θ−1) dθ 

1 1 

− − 

2 2 

= θ (1 −θ) 

1 1 

− − 

2 2 

dς = θ (1 −θ) 

dθ, 

1 A History <strong>of</strong> Mathematical Statistics From 1750 to 1930, John Wiley and Sons Inc., 1998, page 277. 

40

1 1 

a− n−a− 2 2 

P( ς|a,n)d ς ∝ θ (1 − θ) dθ 

. 

However, since under the assumptions we can show that P( θ|a,n)dθ ∝ P( ς|a,n)dς , then 

1 1 

a− n−a− 2 2 

a n−a P( ς|a,n)d ς ∝ θ (1 − θ) dθ 

is inconsistent with P( θ|a,n)d θ ∝ θ (1 − θ) dθ 

. 

To Fisher, the use <strong>of</strong> prior densities (not based on known frequencies) implies that nothing can be known 

about the parameters, regardless <strong>of</strong> the amount <strong>of</strong> information available in the observations. What is 

needed, according to Fisher, is a “rational theory <strong>of</strong> learning by experience”. 

It is noted that (for continuous distributions) the likelihood (function) is not a probability, however it is a 

measure <strong>of</strong> “rational belief”. He writes: 

“Knowing the population we can express our incomplete 

knowledge <strong>of</strong>, or expectation <strong>of</strong>, the sample in terms <strong>of</strong> 

probability; knowing the sample we can express our 

incomplete knowledge <strong>of</strong> the population in terms <strong>of</strong> 

likelihood.” 

Next the concept <strong>of</strong> fiducial distribution is introduced, which is an example <strong>of</strong> the confidence concept 

described by H.A. David and A.W.F. Edwards as: 

Fisher writes: 

“…the idea that a probability statement may be made 

about an unknown parameter (such as limits between 

which it lies, or a value which it exceeds) will be 

correct under repeated sampling from the same 

population.” 1 

“In many cases the random sampling distribution <strong>of</strong> a statistic, T, 

calculable directly from the observations, is expressible solely in terms 

<strong>of</strong> a single parameter, <strong>of</strong> which T is the estimate found by the method <strong>of</strong> 

maximum likelihood. If T is a statistic <strong>of</strong> continuous variation, and P the 

probability that T should be less than any specified value, we have then a 

relation <strong>of</strong> the form 

P =F(T,θ) 

If we now give to P any particular value such as .95, we have a 

relationship between the statistic T and the parameter θ, such that T is the 

95 per cent. Value corresponding to a given θ…” 

In Principles <strong>of</strong> Statistics, by M.G. Bulmer 2, an illustration from a 1935 paper by Fisher is described, with 

sampling from a normal distribution. If a sample <strong>of</strong> size n is taken, the quantity 

1 Annotated Readings in the History <strong>of</strong> Statistics, page 187. 

2 Principles <strong>of</strong> Statistics by M.G. Bulmer, Dover Publications, Inc., New York 1979, page 177. 

41

x 

− µ 

s/ n 

(with notation as usually defined in contemporary statistics) follows a t distribution with n-1 degrees <strong>of</strong> 

freedom. Then 100P per cent <strong>of</strong> those values would be expected to be less than tp, or the probability that 

x − µ 

≤ t 

s/ n 

is equal to P. Fisher notes that the above inequality is equivalent to 

42 

P 

µ ≥ x −st 

/ n 

and reasons that the probability that µ ≥ x − stP/ n is also P. In this case, µ is a random variable and 

x and s are constants. By varying tP, the probability that µ is greater than specific values may be 

obtained, establishing a fiducial distribution for µ, from which fiducial intervals may be constructed. 

Such intervals would correspond to the confidence intervals (as defined in contemporary statistics). 

The interpretation however, is different 1 . In his paper, Fisher provides a table, associated with correlations 

derived from four pairs <strong>of</strong> observations. 

H.A. David and A.W.F. Edwards suggest that Inverse Probability is the first paper clearly identifying the 

confidence concept (although similar approximate constructs such as “probable error” had been in use for 

some time). It is also suggested that Student (W.S. Gosset) first expressed the notion (in an exact way) 

remarking in his 1908 paper: 

“…if two observations have been made and we have no other 

information, it is an even chance that the mean <strong>of</strong> the (normal) 

population will lie between them.” 2 

1 In a confidence interval, µ is a constant, with a certain probability <strong>of</strong> being contained in a random interval. 

2 Annotated Readings…page 187. 

P

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