Mathematical_Recreations-Kraitchik-2e

Probabilities
un
partments, so that the bank has a small constant advantage.
In addition, whenever the ball falls into the 0, all even bets
are withheld and put in play on the next turn. In America
there is usually a second 0 (sometimes even a third), which
probably represents the price of "protection."
Gamblers have expended great ingenuity in devising
schemes for losing their money, and roulette has always been
one of the most prolific sources of such" winning systems."
Here is one.
Suppose the player wishes to ensure winning an average
of 1 unit at each play. For the sake of simplicity let us suppose
that he plays always one of the combinations, such as
red, black, even, or odd, which returns 1 for 1 to the winner.
He begins by betting 1 unit. If he wins the round is closed,
and he again bets 1 unit. If he loses, he bets enough so that
if he wins he gets back his initial loss and 2 units more. If he
wins the increased bet, this round is closed and he again bets
1 unit. If he loses, he must bet enough to cover his losses
and make 3 units more, and so on. Let Xl, X2, ••• , X .. be the
amounts of the successive bets necessary to complete a round
of n plays, that is to ensure winning just n units in n plays of
which the first n - 1 are losses and the nth is a win. Then
we must have:
Xl = 1, - Xl .- X2 + Xa = 3,
- Xl - ~ - ••• - X .. -I t- X .. = n.
Hence
Xl = 1, Xz = 3, Xa = 7, X" = 2" - 1.
Again, if the player always bets on a play which returns 2
for 1, the stakes in the successive plays of a successful round
of n plays are determined by the following equations:
2XI = 1, - Xl + 2X2 = 2, ... , - Xl - Xz - ••• - X .. -I + 2x" = n.
From these we find