Martin has just heard about the following exciting gambling

Martin has just heard about the following exciting gambling strategy: bet $1 that a fair coin will land Heads. If it does, stop. If it lands Tails, double the bet for the next toss, now betting $2 on Heads. If it does, stop. Otherwise, double the bet for the next toss to $4. Continue in this way, doubling the bet each time and then stopping right after winning a bet. Assume that each individual bet is fair, i.e., has an expected net winnings of 0. The idea is that 1+2+2^2+2^3+...+2^n=2^(n+1)-1 so the gambler will be $1 ahead after winning a bet, and then can walk away with a profit. Martin decides to try out this strategy. However, he only has $31, so he may end up walking away bankrupt rather than continuing to double his bet. On average, how much money will Martin win?

Solution

$31 = $1+$2+$4+$8+$16

Thus if Martin keeps loosing he can only go on till 5th bet.

Expected win = P(win on first bet)(gains on first bet)+P(loses first bet but wins second)(gains on 2nd bet)+..... +P(loses first 4 bets but wins fifth)(gains on 5th bet)

=(0.5)($1)+(0.5)(0.5)($2)+(0.5)(0.5)(0.5)($4)+(0.5)(0.5)(0.5)(0.5)($8)+(0.5)(0.5)(0.5)(0.5)(0.5)($16)

=$2.5

Thus the expected net winnings of martin will be $2.5