"It involved tossing a coin until it landed on heads for the first time. In this idea, the more tosses you perform, the more you will win: the proceeds are always doubled. So the first time you get tails, you get $1, and on subsequent times, you get $2, then $4, and so on. Imagine that I offer to play this game with youand I ask for an extremely high stake of $2,000. Will you accept?"
A fair die game pays $10 when the result is 1 or 2, pays $20 when the result is 3, and pays nothing otherwise. The probability of winning $10 is 2/6, and the probability of winning $20 is 1/6. The expected payout is computed by multiplying each winning amount by its probability and summing, giving 40/6, which equals 20/3 dollars, or about $6.66. If a $10 stake is required per roll, the expected net gain becomes 20/3 minus 10, which equals -10/3, or about -$3.33. A related coin-toss game is then introduced where payouts double after each tails until the first heads appears, with a very large stake proposed.
Read at www.scientificamerican.com
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