and therefore the probability that A and B will agree in truly asserting that a white ball is drawn is The probability that a black ball will really be drawn in any ; and therefore the probability that A and B will agree in 9 1 1 falsely asserting that a white ball is drawn is X 10 46° Hence, as in Art. 406, the required probability is Ex. 4. A speaks truth three times out of four, and B five times out of six; and they agree in stating that a white ball has been drawn from a bag which was known to contain 10 balls all of different colours, white being one. What is the chance that a white ball was really drawn? 1 10' The probability that the white ball will really be drawn in any case is and therefore the probability that A and B will agree in 1 3 5 1 10 4 6 16' truly asserting that the white ball is drawn is X X The probability that the white ball will not be drawn in any case 9 is The probability that A will make a wrong statement is 10° hence, as there are nine ways of making a wrong statement which may all be supposed to be equally likely, the chance that A will 1 1 wrongly assert that a white ball is drawn is X Therefore the 49 chance that A and B will agree in falsely asserting that a white ball is drawn is Ex. 5. It is 3 to 1 that A speaks truth, 4 to 1 that B does and 6 to 1 that C does find the probability that an event really took place which A and B assert to have happened and which C denies; the event being, independently of this evidence, as likely to have happened as not. Ans. . 408. We shall conclude this chapter by considering the following examples, referring the reader who wishes for fuller information on the subject of Probabilities to the article in the Encyclopaedia Britannica, and to Todhunter's History of the Mathematical Theory of Probability. Ex. 1. A bag contains n balls, and all numbers of white balls from 0 to n are equally likely; find the chance that r white balls in succession will be drawn, the balls not being replaced. 1 The chance that the bag contains s white balls is ; and the chance that r balls in succession will be drawn from a bag contain If it be known that r white balls in succession have been drawn, the probability of the next drawing giving a white ball can be at once found from the preceding result. For in a great number N, of cases, there will be r white balls in N N succession in cases, and r+1 white balls in succession in r+1 r+2 cases. N N Ex. 2. Two men A and B, who have a and b counters respectively to begin with, play a match consisting of separate games, none of which can be drawn, and the winner of a game receives a counter from the loser. Find their respective chances of winning the match, which is supposed to be continued until one of the players has no more counters, the odds being p: q that A wins any particular game. Let A's chance of ultimate success when he has n counters be un Then A's chance of winning the next game is Ρ and his chance p + q' of ultimate success will then be un+1; also A's chance of losing the •'• pun+1−(p+q) un+qun−1=0, from which it follows that un will be the coefficient of x" in the expansion of provided A and B be properly chosen. A+ Bx But it is obvious that A's chance of winning is zero if he has no counters and unity if he has a+b, so that u=0 and ua+b=1; hence 1. A and B throw alternately with two dice, and a prize is to be won by the one who first throws 8. Find their respective chances of winning if A throws first. 2. A, B and C throw alternately with three dice, and a prize is to be won by the one who first throws 6. Find their respective chances of winning if they throw in the order A, B, C. 3. Three white balls and five black are placed in a bag, and three men draw a ball in succession (the balls drawn not being replaced) until a white ball is drawn: shew that their respective chances are as 27: 18: 11. 4. What is the most likely number of sixes in 50 throws of a die? 5. Shew that with two dice the chance of throwing more than 7 is equal to the chance of throwing less than 7. 6. In a bag there are three tickets numbered 1, 2, 3. A ticket is drawn at random and put back; and this is done four times shew that it is 41 to 40 that the sum of the numbers drawn is even. 7. From a bag containing 100 tickets numbered 1, 2, 3,... 100, two tickets are drawn at random; shew that it is 50 to 49 that the sum of the numbers on the tickets will be odd. 8. There are n tickets in a bag numbered 1, 2, n. A man draws two tickets together at random, and is to receive a number of shillings equal to the product of the numbers he draws: find the value of his expectation. 9. An event is known to have happened n times in n years: shew that the chance that it did not happen in a particular year is ( 1 (1 n n 10. If p things be distributed at random among p persons; shew that the chance that one at least of the persons will be void is p" - p 11. A writes a letter to B and does not get an answer; assuming that one letter in m is lost in passing through the post, shew that the chance that B received the letter is - 1 m 2m - 1' it being considered certain that B would have answered the letter if he had received it. 12. From a bag containing 3 sovereigns and 3 shillings, four coins are drawn at random and placed in a purse; two coins are then drawn out of the purse and found to be both sovereigns. Shew that the value of the expectation of the remaining coins in the purse is 11s. 6d. 13. From a bag containing 4 sovereigns and 4 shillings, four coins are drawn at random and placed in a purse; two coins are then drawn out of the purse and found to be both sovereigns. Shew that the probable value of the coins left in the bag is 29 shillings. 14. If three points are taken at random on a circle the chance of their lying on the same semi-circle is 3. 15. A rod is broken at random into three pieces: find the chance that no one of the pieces is greater than the sum of the other two. 16. A rod is broken at random into four pieces: find the chance that no one of the pieces is greater than the sum of the other three. 17. Three of the sides of a regular polygon of 4n sides are chosen at random; prove that the chance that they being produced will form an acute-angled triangle which will contain (n-1) (n-2) the polygon is (4n-1) (4n-2) 18. Out of m persons who are sitting in a circle three are selected at random; prove that the chance that no two of (m-4) (m-5) those selected are sitting next one another is (m − 1) (m − 2) * 19. If m odd integers and n even integers be written down at random, shew that the chance that no two odd numbers are adjacent to one another is 20. If m things are n n + 1 women, shew that the chance that the number of things received by the group of men is odd, is 1 (b+a) (ba)m 2 (b + a)m |