Mathematical_Recreations-Kraitchik-2e

CHAPTER SIX
PROBABILITIES
THE serious investigation of the laws of chance began with
Pascal and Fermat in the seventeenth century. Our first
problem is a more general statement of a question brought
to Pascal by the Chevalier de Mere, a gambler friend of
Pascal. It was this problem which started Pascal on his researches
in the field of probability.
1. THE UNFINISHED GAME. A and B are playing a game
in which a point is scored for either A or B at the end of each
play, and the first to win N points wins the game. Thus there
are no drawn games. The game is interrupted when A needs
a points in order to win, and B needs b points. How should
the stakes be divided between the players?
Solution: Let m and n = 1 - m be the respective probabilities
that A and B will win anyone point, and p and q =
1 - p their respective chances of winning the game at the
time the game stopped. Then the stakes should be divided
in the ratio p: q, where
= a[l+~ +a(a+1) 2+ ... + a(a+1) ... (a+b-2) b-1J
p mIn 1 . 2 n (b _ I)! n,
= b[l+!!' +b(b+I) 2+ ... + b(b+1) ... (b+a-2) .. -lJ
q n 1 mI. 2 m (a _ I)! m.
If the game is one in which m = n = !, as in matching pennies
or cutting for the high card, then the relative values of
p and q may be listed in such a table as the following:
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