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# A gambling problem

## Question

A gambler has \$5,000 and is playing a game of chance with a win probability of .95. Every time he wins, he raises his stake to 1/4, of his bankroll. The gambler doesn't reduce his stake when he loses

If he keeps at it, what are his expected winnings?

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Just at a glance,

Spoiler

The gambler goes bankrupt if he loses four times in a row. That occurs with some non-zero probability, so if he keeps going on forever, he'll eventually go bankrupt.

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1. What is his initial bet (stake)?
2. Did you mean to say "Every time he wins, he raises his stake by 1/4 of his bankroll"?
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6 hours ago, bonanova said:
1. What is his initial bet (stake)?
2. Did you mean to say "Every time he wins, he raises his stake by 1/4 of his bankroll"?

His initial bet is \$1250.  Every time he wins, he calculates his total and takes 1/4 of it to bet.  So initially he has 5000, 5000 x .25 = 1250.  If he were to win, he would earn double his bet so he would have \$6250.  Recalculating he would bet \$1562.50.  If he loses his bet he would maintain that amount in his next bet.  He keeps at it until he is unable to make his desired bet.

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