45. In a bag are m white and n black balls. Shew that the chance of drawing first a white, then a black ball, and so on alternately until the balls remaining are all of one colour is

mn m + n

If m balls are drawn at once, what is the chance of drawing all the white balls at the first trial?

...

46. In a bag are n balls of m colours, p, being of the first colour, p, of the second colour, Pm of the mth colour. If the balls be drawn one by one, what is the chance that all the balls of the first colour will be first drawn, then all the balls of the second colour, and so on, and lastly all the balls of the mth colour?

47. A bag contains n balls; a person takes out one and puts it in again; he does this n times; what is the probability of his having had in his hand every ball in the bag?

48. Two players of equal skill, A and B, are playing a set of games. A wants 2 games to complete the set, and B wants 3 games. Compare the chances of A and B for winning the set.

49. If three persons dine together, in how many different ways can they be seated? When they have dined together exactly so many times, taking their places by chance, what is the probability that they will have sat in every possible arrangement?

50. N is a given number; a lower number is selected at random, find the chance that it will divide N.

51.

A handful of shot is taken at random out of a bag; what is the chance that the number of shot in the handful is prime to the number of shot in the bag? For example, suppose the number of shot in the bag to be 105.

52. If n=a", and any number not greater than ʼn be taken at random, the chance that it contains a as a factor s times and no

53. Two persons play at a game which cannot be drawn, and agree to continue to play until one or other of them wins two games in succession; given the chance that one of them wins a single game, find the chance that he wins the match described. For example, if the odds on a single game be 2 to 1, the odds on the match will be 16 to 5.

54. A person has a pair of dice, one a regular tetrahedron, the other a regular octahedron; what is the chance that in a single throw the sum of the marks is greater than 6?

55. There are three independent events of which the probabilities are respectively P1, P2, P3; find the probability of the happening of one of the events at least; also of the happening of two of the events at least.

56. A certain sum of money is to be given to one of three persons A, B, C, who first throws 10 with three dice; supposing them to throw successively in the order named until the event has happened, shew that their chances are respectively

57. The decimal parts of the logarithms of two numbers taken at random are found from a table to 7 places; what is the probability that the second can be subtracted from the first without borrowing at all?

58. A undertakes with a pair of dice to throw 6 before B throws 7; they throw alternately, A commencing. Compare their chances.

59.

A person is allowed to draw two coins from a bag containing 4 sovereigns and 4 shillings. What is the value of his expectation?

60. If six guineas, six sovereigns, and six shillings be put into a bag, and three be drawn out at random, what is the value of the expectation?

61. Ten Russian ships, twelve French, and fourteen English are expected in port. What is the value of the expectation of a merchant who will gain £2100 if one of the first two which arrive is a Russian and the other a French ship?

62. From a bag containing 3 guineas, 2 sovereigns, and 4 shillings, a person draws 3 coins indiscriminately; what is the value of his expectation?

63. What is the worth of a lottery-ticket in a lottery of 100 tickets, having four prizes of £100, ten of £50, and twenty of £5 ?

64. A bag contains 9 coins, 5 are sovereigns, the other four are equal to each other in value; find what this value must be in order that the expectation of receiving two coins out of the bag may be worth 24 shillings.

65. From a bag containing 4 shilling pieces, 3 unknown silver coins of the same value, and one unknown gold coin, four are to be drawn. If the value of the drawer's chance be 15 shillings, what are the coins?

66. A and B subscribe a sum of money for which they toss alternately beginning with A, and the first who throws a head is to win the whole. In what proportion ought they to subscribe? If they subscribe equally, how much should either of them give the other for the first throw ?

67. There are a number of counters in a bag of which one is marked 1, two 2, &c. up to r marked r; a person draws a number at random for which he is to receive as many shillings as the number marked on it; find the value of his expectation.

68. A bag contains a number of tickets of which one is marked 1, four marked 2, nine marked 3,

...

up to n2 marked n;

a person draws a ticket at random for which he is to receive as many shillings as the number marked on it; required the value of his expectation.

69. A man is to receive a certain number of shillings, he knows that the digits of the number are 1, 2, 3, 4, 5, but he is ignorant of the order in which they stand; determine the value of his expectation.

70. From a bag containing a counters some of which are marked with numbers, b counters are to be drawn, and the drawer is to receive a number of shillings equal to the sum of the numbers on the counters which he draws; if the sum of the numbers on all the counters be n, what will be the value of his chance?

71. There are two urns, and it is known that one contains 8 white balls and 4 black balls, and that the other contains 12 black balls and 4 white balls; from one of these, but it is not known from which, a ball is taken and is found to be white; find the chance that it was drawn from the urn containing 8 white balls.

72. Five balls, any one of which may be either white or black, are in a bag, and two being drawn are both white; find the probability that all are white.

73. A purse contains n coins which are either sovereigns or shillings; a coin drawn is a sovereign, what is the probability that this is the only sovereign?

74. A bag contains 4 white and 4 red balls; two are taken out at random, and without being seen are placed in a smaller bag; one is taken out and proves to be white, and replaced in the smaller bag; one is again taken out and proves to be again white, what is now the probability that both balls in the smaller bag are white?

75. Of two purses one originally contained 25 sovereigns, and the other 10 sovereigns and 15 shillings. One purse is taken by chance and 4 coins drawn out which prove to be all sovereigns; what is the probability that this purse contains only sovereigns, and what is the value of the expectation of the next coin that will be drawn from it?

76. A bag contains three bank notes, and it is known that each of them is either a £5, a £10, or a £20 note; at three successive dips in the bag (the note being replaced after each dip) a £5 note was drawn. What is the probable value of the contents of the bag?

77. It is 3 to 1 that A speaks the truth, 4 to 1 that B does, and 6 to 1 that C does; what is the probability that an event took place which A and B assert to have happened and which C denies ?

78. A speaks truth 3 times out of 4, B 4 times out of 5; they agree in asserting that from a bag containing 9 balls, all of different colours, a white ball has been drawn; shew that the proba96 97

bility that this is true is

79. Suppose thirteen witnesses, each of whom makes but one false statement in eleven, to assert that a certain event took place ; shew that the odds are ten to one in favour of the truth of their statement, even although the a priori probability of the event be

80. One of a pack of 52 cards has been removed; from the remainder of the pack two cards are drawn and are found to be spades; find the chance that the missing card is a spade.

81. If two persons walk on the same road in opposite directions during the same interval of time a+b+c, the one completing the distance in a time a, and the other in a time b, what is the chance of their meeting?

82. Find how many odd numbers taken at random must be multiplied together, that there may be at least an even chance of the last figure being 5.

Given log12 = 30103.

LIV.

MISCELLANEOUS EQUATIONS.

747. Equations may be proposed which require peculiar artifices for their solution; in the following collection the student will find ample exercise; he should himself try to solve the equations, and afterwards consult the solution here given. x2+4x+6 x2+6x+12

x2 + 2x + 2

x2 + 8x+20

1.

+

=

x + 1

+

x+3