Mathematical_Recreations-Kraitchik-2e

140 Mathematical Recreations
0.391 for B. In order to have an equitable game the stakes
must be in this ratio, 609:391.
20. THE GAMBLER'S RUIN. If two persons play constantly
against each other, one of them will end by ruining
the other, whether the game be fair or not. The probability
that they can play a given number n of games approaches
° as n increases. For example, if one plays heads or tails at
the rate of a dollar a game, he has a probability bordering on
2.1016
certainty (actually 0.999) of losing $100,000 if he plays --
7f"
games; and a probability of 0.9 of losing $100 in 624,000
games.
Though these results seem reassuring, ruin is certain.
Furthermore, it was assumed that the accounts were not settled
until the end of the game. Under ordinary conditions
one must put up his stake before each game, so it is not the
ultimate loss which should be considered, but the maximum.
In an equitable game two players whose fortunes are a and
b have the respective probabilities a : band a ! b of ruining
each other. If the game is not equitable, let p and q be the
respective probabilities for each to win a game. Then the
probability that the first player will ruin the second is X~b-_\,
where x is found from the equation px<>+b - X" + q = 0.
If the game is not equitable the least advantage is sufficient
not merely to avoid ruin but to ensure a profit. This is the
case in roulette, where the banker has the advantage of the 0.
It is also the case with insurance companies.
21. A man stakes ~ of his fortune on each play of a game
m
whose probability is!. He tries again and again, each time