Thus, if n = 3, raising a b to the cube + 3 a' b,-+ 3 a b' b3, all the terms but b3 will be favourable to A; and therefore the probability of A's • • be or winning wi a +613 a 3 — 63 ; and the probability of B's M3 b 3 winning will be But if A and B a play on condition that, if either two or more of the events in question happen, A shall win; but in case one only happen, or none, B shall win; the probability of A's a -775 — n b.-1— b.; winning will be n bn for the only two terms in which a a does not occur are the two last, 7'lZ. n a b and be. See Simpson's "Nature and Laws of Chance." We shall now add a table that may be useful to persons not skilled in mathematics, and which is ap plicable to many subjects: The above proportions are found by the binomial theorem in a very easy way. Suppose the games wanting 1 and 5, raise a+ b to the fifth power, being the number of games which must determine the bet. a=b in this case, as the skill is equal : as+5, a4 6+ 19, a3 b' + 10, a' 63 + 5, a 64 + 63, the first five coefficients are the chances of him who has 1 game to get, viz. 1+ 5+ 10+ 10 + 5 ,-.- 31, and the other, viz. 1, the chance of him who has five to get.
Suppose the games wanting are 2 and 5, then aS 6 4-15, a+ b' ± 20, 0363 + 15, a' + 6, a bs + the chances for him wanting two are 1 + 6 + 15 + 0+ 15 = 57; but for him wanting 5, re 6 + 1= 7 according to table 57 : 7.
Suppose the games wanting 4 and 6, en a9+9, as b' + 84, 26, as 64 + 126, a4 65+ 84, a3 + 36, 4 67 + 9, a therefore for him wanting 4 games, 1 + 9 + 36 + 84 + 126 + 126 = 382, and to him wanting 6 re 84 + 36 + 9 + 1 = 130: the odds *re 382: 130 according to table.
0 When the skill is not equal, or when the chances for winning are not equal: as, ' 1. If A and B play together, and A ants 1 game of being up, and B wants ; but the chances whereby B may win A game are double to the number of -chances whereby A may win the same. Here the number of games are two. And fa = 1 and b = 2 .-. + 2 a b + 6' will give the probability of each. A ._---- 1 + 4 = 5 and B = 4 or the probabilities are A • B:: 5 : 4.
2. A wants 3 games of being up, B .7; the proportion of chances 3 to 5, what is the proportion of chances to win the set? here the number of games will be 9, a = 3 6 = 5, therefore raise and the three last terms + by a + 69 will express the chances of B, which sub