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Topic: Coin Flip Game Worth IV (Read 1008 times) |
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rugga
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Coin Flip Game Worth IV
« on: Sep 7th, 2002, 10:37am » |
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Here's a puzzle I heard a while back rephrased as a cross between coin flip games II and III:
Game: You continue flipping a coin until you get a tails. I then award you prize money of $1 for every flip performed.
How much are you willing to pay to play this game?
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| « Last Edit: Oct 23rd, 2003, 7:43pm by Icarus » |
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Pietro K.C.
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Re: NEW PROBLEM: Coin Flip Game Worth IV
« Reply #1 on: Sep 7th, 2002, 8:13pm » |
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I suppose I would pay up to the expected winnings, if I can properly calculate them. Let they be represented by W. There is 1/2 chance that I will get $1 (tails turns up first), 1/4 that I get $2, 1/8 that I get $3, etc, so that
W = sum i / 2^i. (i = 1 to oo)
To avoid tiresome calculation of limits (which CAN be done rigorously, but let us pretend to be clever), we note that
W = 1/2 + sum i / 2^i (i = 2 to oo)
therefore
W = 1/2 + sum (i+1) / 2^(i+1) (i = 1 to oo)
and hence
W = 1/2 + 1/2 ( sum i / 2^i ) + 1/2 ( sum 1 / 2^i ). (i = 1 to oo in both sums)
So we have
W = 1/2 + W/2 + 1/2 (the rightmost sum is just 1/2 + 1/4 +...)
Therefore W = 2.
So, mathematically, I would pay you at most two bucks to play this game. Realistically though, I would go play poker instead, which at least is fun. 
Let me know if there is something else to this problem that I didn't get.
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| « Last Edit: Sep 7th, 2002, 8:20pm by Pietro K.C. » |
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"I always wondered about the meaning of life. So I looked it up in the dictionary under 'L' and there it was --- the meaning of life. It was not what I expected." (Dogbert)
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rugga
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Re: NEW PROBLEM: Coin Flip Game Worth IV
« Reply #2 on: Sep 9th, 2002, 2:07am » |
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That's the right answer. But I agree about poker 
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