A gambling problem

[BMAD]

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?

[gavinksong]

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.

 

[bonanova]

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"?

[BMAD]

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.

Edited September 11, 2017 by BMAD