aez-notes

At Las Vegas, a man with \(\$20\) needs \(\$40\), but he is too embarrassed to wire his wife for more money. He decides to invest in roulette (which he doesn't enjoy playing) and is considering two strategies: bet the \(\$20\) on "evens" all at once and quit if he wins or loses, or bet on "evens" one dollar at a time until he has won or lost \(\$20\). Compare the merits of the strategies.

The expected return in a bet on "evens" is negative, so the man has a better chance of winning if he bets it all at once. To see why, consider the case of making smaller and smaller bets. In the limit he will certainly loose and the chance is monotonic in the number of bets.