Flip until tails, earn two dollars per flip

Question

You flip a fair coin repeatedly until the first time it lands tails, then you stop. You earn 2 for every flip you make, including the final tails. On average, how much do you earn?

Accepted Answer

Earnings are 2 per flip, so find the average number of flips first, then multiply. Average number of flips. A fair coin lands tails with probability one half on each flip. There is a general rule: if an event has probability p on each independent try, the average number of tries until it first happens is 1/p. Here the target event is tails with p = 1/2, so the average number of flips is: \frac{1}{1/2} = 2. If you want to see why, note you always make one flip, and half the time it is heads and you start over. Writing average flips as N = 1 + 	frac{1}{2}N and solving gives N = 2 as well. Aver…

Flip until tails, earn two dollars per flip

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