Consider betting on red or black at the roulette table. The payout is even. (That means if you bet $1 and win, you win $1. But if you lose, you lose your $1.) And, for simplicity, assume that you really have a 50–50 shot of calling the correct color. (Real roulette tables have some additional green pockets on which you lose, giving the house a slight edge.) We’ll also suppose that the table has no maximum bet.
A disconcerting 28 percent of participants went broke despite having an advantage, and a shocking two thirds bet on tails at some point in the game, which is never rational. On average, the participants walked away with $91 (winnings were capped at $250). This might seem like an ample take for someone starting with $25, but the researchers calculated that over the 300 coin tosses time allowed for, the average winnings of players using the optimal strategy (described below) would be more than $3 million!
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Do you spot a flaw in the profit-promising reasoning above? Say you have $7 in your pocket, and you’d like to turn it into $8. You can afford to lose the first three bets in a row of $1, $2 and $4. It’s not very likely that you will lose three in a row, though, because the probability is only one in eight. So one eighth (or 12.5 percent) of the time you’ll lose all $7, and the remaining seven eighths of the time you’ll gain $1. These outcomes cancel each other out: −1/8 × $7 + 7/8 × $1 = $0.
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