Flip until heads for twenty dollars, minus three per flip

Question

You flip a fair coin until the first heads, then stop and collect a flat 20 prize. But each flip costs you 3, including the flip that lands heads. On average, how much do you net from this game?

Accepted Answer

There are two parts: a fixed 20 prize you always collect, and a per-flip cost that depends on how many flips you make. Handle each. The prize. You always reach a first heads eventually, so you always collect the 20. That part contributes a flat 20 on average. The flip cost. Each flip costs 3, so the total cost is 3 times the number of flips. The average number of flips until a fair coin's first heads is 1/p with p = 1/2, which is 2. So the average cost is: 3 	imes 2 = 6. Net. Subtract the average cost from the prize: EV = 20 - 6 = 14. You net 14 on average. The prize is fixed, but the cost i…

Flip until heads for twenty dollars, minus three per flip

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