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Flipping Quarters

Here's an interesting puzzle involving chance:

A man in a park asks you to play a game with him. It's a form of gambling. To play, you must pay the man $5, then flip a coin repeatedly until you get heads. As soon as you get heads, you stop flipping. If you only flipped the quarter once, he'll give you $1. If you flipped it twice, you get $2. Three times, $4. Four times, $8. Each extra flip gets you twice as much money, so the longer it takes before you get tails, the more money you get.

Should you play, if you have a lot of time and the man will play as many games as you want? How much money, on average, would you gain (subtracting the $5 fee)?

I will give the solution in a later post.

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  1. Dave ShanklinDecember 31, 2015 at 7:28 PM

    1. As long as you can quit and he can't.
    2. Have a substantial bankrole (assets to risk).
    3. -5 + (1/2)(0) + (1/4)(1) + .(1/8)(2) + (1/16)(4) + 1/32(8) + 1/64(16) + .....=
    = -5 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 .... = -5 + (n-1)*(1/4) -> which goes to infinity as the n approaches infinity.
    4. Reasons to quit.
    a. (bankroll is exhausted or there is a small risk of flipping tails a large number of times); Check the coin, if this seems to be the case.
    b. Can make a better ROI doing another activity either in dollars or personal satisfaction.
    c. Moral compulsion of not taking advantage of someone else.

    Note: There are limits to betting on tables in vegas (this is generally to protect the house) :)

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