one; in fact it often amounts to nothing more than a distinction between two different modes of viewing the same occurrence.

Example. A bag contains 5 white and 7 black balls; if two balls are drawn what is the chance that one is white and the other black?

(i) Regarding the occurrence as a simple event, the chance

(ii) The occurrence may be regarded as the happening of one or other of the two following compound events:

(1) drawing a white and then a black ball, the chance of which is

(2) drawing a black and then a white ball, the chance of which is

And since these events are mutually exclusive, the required chance

It will be noticed that we have here assumed that the chance of drawing two specified balls successively is the same as if they were drawn simultaneously. A little consideration will shew that this must be the case.

EXAMPLES. XXXII. b.

1. What is the chance of throwing an ace in the first only of two successive throws with an ordinary die?

2. Three cards are drawn at random from an ordinary pack: find the chance that they will consist of a knave, a queen, and a king.

3. The odds against a certain event are 5 to 2, and the odds in favour of another event independent of the former are 6 to 5: find the chance that one at least of the events will happen.

4. The odds against A solving a certain problem are 4 to 3, and the odds in favour of B solving the same problem are 7 to 5: what is the chance that the problem will be solved if they both try?

5. What is the chance of drawing a sovereign from a purse one compartment of which contains 3 shillings and 2 sovereigns, and the other 2 sovereigns and 1 shilling?

6. A bag contains 17 counters marked with the numbers 1 to 17. A counter is drawn and replaced; a second drawing is then made: what is the chance that the first number drawn is even and the second odd?

7. Four persons draw each a card from an ordinary pack: find the chance (1) that a card is of each suit, (2) that no two cards are of equal value.

8. Find the chance of throwing six with a single die at least once in five trials.

9. The odds that a book will be favourably reviewed by three independent critics are 5 to 2, 4 to 3, and 3 to 4 respectively; what is the probability that of the three reviews a majority will be favourable?

10. A bag contains 5 white and 3 black balls, and 4 are successively drawn out and not replaced; what is the chance that they are alternately of different colours?

11. In three throws with a pair of dice, find the chance of throwing doublets at least once.

12. If 4 whole numbers taken at random are multiplied together shew that the chance that the last digit in the product is 1, 3, 7, or 9

13. In a purse are 10 coins, all shillings except one which is a sovereign; in another are ten coins all shillings. Nine coins are taken from the former purse and put into the latter, and then nine coins are taken from the latter and put into the former: find the chance that the sovereign is still in the first purse.

14. If two coins are tossed 5 times, what is the chance that there will be 5 heads and 5 tails?

15. If 8 coins are tossed, what is the chance that one and only one will turn up head?

16. A, B, C in order cut a pack of cards, replacing them after each cut, on condition that the first who cuts a spade shall win a prize: find their respective chances.

17. A and B draw from a purse containing 3 sovereigns and 4 shillings: find their respective chances of first drawing a sovereign, the coins when drawn not being replaced.

18. A party of n persons sit at a round table, find the odds against two specified individuals sitting next to each other.

19. A is one of 6 horses entered for a race, and is to be ridden by one of two jockeys B and C. It is 2 to 1 that B rides A, in which case all the horses are equally likely to win; if C rides A, his chance is trebled what are the odds against his winning?

20. If on an average 1 vessel in every 10 is wrecked, find the chance that out of 5 vessels expected 4 at least will arrive safely.

462.

The probability of the happening of an event in one trial being known, required the probability of its happening once, twice, three times, ... exactly in n trials.

Let p be the probability of the happening of the event in a single trial, and let q=1-p; then the probability that the event will happen exactly r times in n trials is the (r+ 1)th term in the expansion of (q+p)".

For if we select any particular set of r trials out of the total number n, the chance that the event will happen in every one of these r trials and fail in all the rest is p'q" [Art. 456], and as a set of r trials can be selected in "C, ways, all of which are equally applicable to the case in point, the required chance is

If we expand (p + q)" by the Binomial Theorem, we have

n-2

p"+"C,p"-q+"C2p"-q2 + +"C-p'q" + ... + q" ;

...

thus the terms of this series will represent respectively the probabilities of the happening of the event exactly n times, n − 1 times, n 2 times, ... in n trials.

463. If the event happens n times, or fails only once, twice, (n − r) times, it happens r times or more; therefore the chance that it happens at least r times in n trials is

...

n-22

p" +"C1p"-1q + "C ̧p"−2q2 + ... + "C-p'q"-",

or the sum of the first n-r+1 terms of the expansion of (p + q)".

Example 1. In four throws with a pair of dice, what is the chance of throwing doublets twice at least?

6

In a single throw the chance of doublets is or and the chance of

36

failing to throw doublets is Now the required event follows if doublets

5
6'

are thrown four times, three times, or twice; therefore the required chance of the first three terms of the

is the sum

Thus the chance

H. H. A.

expansion of (+5)*.

Example 2. A bag contains a certain number of balls, some of which are white; a ball is drawn and replaced, another is then drawn and replaced; and so on: if p is the chance of drawing a white ball in a single trial, find the number of white balls that is most likely to have been drawn in n trials.

The chance of drawing exactly r white balls is "C,p"q"-", and we have to find for what value of r this expression is greatest.

Now so long as

or

"C„p1qn-r>"Cr-1pr—1qn−(r−1),

(n-r+1)p>rq,

(n+1)p>(p+q)r.

But p+q=1; hence the required value of r is the greatest integer in p (n+1).

If n is such that pn is an integer, the most likely case is that of pn successes and qn failures.

464. Suppose that there are n tickets in a lottery for a prize of £x; then since each ticket is equally likely to win the prize, and a person who possessed all the tickets must win, the money value of each ticket is £

х

n

;

in other words this would be a fair sum to

pay for each ticket; hence a person who possessed r tickets might

reasonably expect £

rx n

as the price to be paid for his tickets by

any one who wished to buy them; that is, he would estimate

£ -x as the worth of his chance. It is convenient then to in

n

troduce the following definition :

If p represents a person's chance of success in any venture and M the sum of money which he will receive in case of success, the sum of money denoted by pM is called his expectation.

465. In the same way that expectation is used in reference to a person, we may conveniently use the phrase probable value applied to things.

Example 1. One purse contains 5 shillings and 1 sovereign: a second purse contains 6 shillings. Two coins are taken from the first and placed in the second; then 2 are taken from the second and placed in the first : find the probable value of the contents of each purse.

The chance that the sovereign is in the first purse is equal to the sum of the chances that it has moved twice and that it has not moved at all;

.. the chance that the sovereign is in the second purse=

.. the probable value of the first purse

= (25 −81+312) shillings = £1. Os. 3d., as before.

Example 2. A and B throw with one die for a stake of £11 which is to be won by the player who first throws 6. If A has the first throw, what are their respective expectations?

each player must have failed once before A can have a second throw; in his

third throw his chance is

twice; and so on.

Similarly B's chance is the sum of the infinite series

.. A's chance is to B's as 6 is to 5; their respective chances are therefore

5

and

11

and their expectations are £6 and £5 respectively.