Biased coin, earn a dollar per flip until heads

Question

You have a biased coin that lands heads only one third of the time (and tails two thirds of the time). You flip it repeatedly until the first heads, then stop. You earn $1 for every flip, including the heads.

On average, how much do you earn?

Accepted Answer

Earnings are 1 per flip, so the answer is just the average number of flips. Average number of flips. The rule for waiting time: if an event has probability p on each independent try, the average number of tries until it first happens is 1/p. Here heads has probability p = 1/3, so the average number of flips is: \frac{1}{1/3} = 3. The intuition: a rarer event takes longer to wait for. A heads that shows up only one time in three needs about three flips on average, compared to two flips for a fair coin. The less likely the event, the longer the wait, in exact proportion. Average earnings. At 1 …

Biased coin, earn a dollar per flip until heads

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