Consider a simple gambling game where you place a bet of $n, flip a fair coin, and win $n if it comes up heads and lose $n if it comes up tails. Now consider the following gambling strategy for playing this game.
- Start by betting $1.
- Each time you lose, bet twice as much the next time.
- As soon as you win, quit.
a) If you win on the first try, how much money do you win?
b) If you lose three times before winning, how much money do you win/lose (net over all the plays)?
c) If you lose k times before winning, how much money do you win/lose (net over all the plays)?
d) Let X be the amount of money bet in the last round (when you win). Determine a formula for P(X = 2k). (The only possible values for X are powers of two because you double the bet each round.)
e) Use the definition of expected value to calculate E(X). What is strange about this? What problem does it reveal with this betting scheme?