466.

We shall now give two problems which lead to useful and interesting results.

Example 1. Two players A and B want respectively m and n points of winning a set of games; their chances of winning a single game are p and q respectively, where the sum of p and q is unity; the stake is to belong to the player who first makes up his set: determine the probabilities in favour of each player.

Suppose that A wins in exactly m+r games; to do this he must win the last game and m-1 out of the preceding m+r-1 games. The chance of this is m+r-1Cm ̧ pm−1 qrp, or m+r−1Cm¬1 pm q”..

m-1

Now the set will necessarily be decided in m+n-1 games, and A may win his m games in exactly m games, or m+1 games, or m+n-1 games; therefore we shall obtain the chance that A wins the set by giving to r the values 0, 1, 2, ...n-1 in the expression m+r-1Cm-1pm q". Thus A's chance is

This question is known as the "Problem of Points," and has engaged the attention of many of the most eminent mathematicians since the time of Pascal. It was originally proposed to Pascal by the Chevalier de Méré in 1654, and was discussed by Pascal and Fermat, but they confined themselves to the case in which the players were supposed to be of equal skill: their results were also exhibited in a different form. The formule we have given are assigned to Montmort, as they appear for the first time in a work of his published in 1714. The same result was afterwards obtained in different ways by Lagrange and Laplace, and by the latter the problem was treated very fully under various modifications.

Example 2. There are n dice with ƒ faces marked from 1 to f; if these are thrown at random, what is the chance that the sum of the numbers exhibited shall be equal to p?

Since any one of the f faces may be exposed on any one of the n dice, the number of ways in which the dice may fall is ƒ”.

Also the number of ways in which the numbers thrown will have their sum is equal to the coefficient of x" in the expansion of

(x2+x2+x3+... +x3)n;

p for

for this coefficient arises out of the different ways in which n of the indices 1, 2, 3, ... ƒ can be taken so as to form p by addition.

and

Now the above expression = (1+x+x2+...+xs−1)n

We have therefore to find the coefficient of xp-" in the expansion of

Now (1-x) =1-nướ

Multiply these series together and pick out the coefficient of x-" in the product; we thus obtain

where the series is to continue so long as no negative factors appear. required probability is obtained by dividing this series by f".

The

This problem is due to De Moivre and was published by him in 1730; it illustrates a method of frequent utility.

Laplace afterwards obtained the same formula, but in a much more laborious manner; he applied it in an attempt to demonstrate the existence of a primitive cause which has made the planets to move in orbits close to the ecliptic, and in the same direction as the earth round the sun. On this point the reader may consult Todhunter's History of Probability, Art. 987.

EXAMPLES. XXXII. c.

1. In a certain game A's skill is to B's as 3 to 2: find the chance of A winning 3 games at least out of 5.

2. A coin whose faces are marked 2, 3 is thrown 5 times: what is the chance of obtaining a total of 12?

3. In each of a set of games it is 2 to 1 in favour of the winner of the previous game: what is the chance that the player who wins the first game shall win three at least of the next four?

4. There are 9 coins in a bag, 5 of which are sovereigns and the rest are unknown coins of equal value; find what they must be if the probable value of a draw is 12 shillings.

5. A coin is tossed n times, what is the chance that the head will present itself an odd number of times?

6. From a bag containing 2 sovereigns and 3 shillings a person is allowed to draw 2 coins indiscriminately; find the value of his expectation.

7. Six persons throw for a stake, which is to be won by the one who first throws head with a penny; if they throw in succession, find the chance of the fourth person.

8. Counters marked 1, 2, 3 are placed in a bag, and one is withdrawn and replaced. The operation being repeated three times, what is the chance of obtaining a total of 6?

9. A coin whose faces are marked 3 and 5 is tossed 4 times: what are the odds against the sum of the numbers thrown being less than 15? 10. Find the chance of throwing 10 exactly in one throw with 3 dice.

11. Two players of equal skill, A and B, are playing a set of games; they leave off playing when A wants 3 points and B wants 2. If the stake is £16, what share ought each to take?

12. A and B throw with 3 dice: if A throws 8, what is B's chance of throwing a higher number?

13. A had in his pocket a sovereign and four shillings; taking out two coins at random he promises to give them to B and C. What is the worth of C's expectation?

14. In five throws with a single die what is the chance of throwing (1) three aces exactly, (2) three aces at least.

15. A makes a bet with B of 5s. to 2s. that in a single throw with two dice he will throw seven before B throws four. Each has a pair of dice and they throw simultaneously until one of them wins: find B's expectation.

16. A person throws two dice, one the common cube, and the other a regular tetrahedron, the number on the lowest face being taken in the case of the tetrahedron; what is the chance that the sum of the numbers thrown is not less than 5?

17. A bag contains a coin of value M, and a number of other coins whose aggregate value is m. A person draws one at a time till he draws the coin M: find the value of his expectation.

18. If 6n tickets numbered 0, 1, 2, ...... 6n-1 are placed in a bag, and three are drawn out, shew that the chance that the sum of the numbers on them is equal to 6n is

*INVERSE PROBABILITY.

*467. In all the cases we have hitherto considered it has been supposed that our knowledge of the causes which may produce a certain event is such as to enable us to determine the chance of the happening of the event. We have now to consider problems of a different character. For example, if it is known that an event has happened in consequence of some one of a certain number of causes, it may be required to estimate the probability of each cause being the true one, and thence to deduce the probability of future events occurring under the operation of the

same causes.

*468. Before discussing the general case we shall give a numerical illustration.

Suppose there are two purses, one containing 5 sovereigns and 3 shillings, the other containing 3 sovereigns and 1 shilling, and suppose that a sovereign has been drawn: it is required to find the chance that it came from the first or second purse.

Consider a very large number N of trials; then, since before the event each of the purses is equally likely to be taken, we may assume that the first purse would be chosen in N of the trials,

12N

and in of these a sovereign would be drawn ; thus a sovereign

5
8

5

would be drawn

8

16

× N, or N times from the first purse.

3

4

1 2

5

1

The second purse would be chosen in N of the trials, and in

2

of these a sovereign would be drawn; thus a sovereign would

be drawn

3

8

N times from the second purse.

Now N is very large but is otherwise an arbitrary number; let us put N16n; thus a sovereign would be drawn 5n times from the first purse, and 6n times from the second purse; that is, out of the 11n times in which a sovereign is drawn it comes from the first purse 5n times, and from the second purse 6n

*469. It is important that the student's attention should be directed to the nature of the assumption that has been made in the preceding article. Thus, to take a particular instance, although in 60 throws with a perfectly symmetrical die it may not happen that ace is thrown exactly 10 times, yet it will doubtless be at once admitted that if the number of throws is continually increased the ratio of the number of aces to the number of throws will tend more and more nearly to the limit There is no reason why one face should appear oftener than 6' another; hence in the long run the number of times that each of the six faces will have appeared will be approximately equal.

1

The above instance is a particular case of a general theorem which is due to James Bernoulli, and was first given in the Ars Conjectandi, published in 1713, eight years after the author's death. Bernoulli's theorem may be enunciated as follows:

If p is the probability that an event happens in a single trial, then if the number of trials is indefinitely increased, it becomes a certainty that the limit of the ratio of the number of successes to the number of trials is equal to p; in other words, if the number of trials is Ñ, the number of successes may be taken to be pN.

See Todhunter's History of Probability, Chapter VII. A proof of Bernoulli's theorem is given in the article Probability in the Encyclopædia Britannica.

*470. An observed event has happened through some one of a number of mutually exclusive causes: required to find the probability of any assigned cause being the true one.

Let there be n causes, and before the event took place suppose that the probability of the existence of these causes was estimated at P1, P, P, ... P. Let p, denote the probability that when the P1, 7th cause exists the event will follow: after the event has occurred it is required to find the probability that the 7th cause was the

true one.