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        riddles >> medium >> Coin Flip Game Worth IV
        
(Message started by: rugga on Sep 7^th, 2002, 10:37am)

Title: Coin Flip Game Worth IV
Post by rugga on Sep 7^th, 2002, 10:37am

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?

Title: Re: NEW PROBLEM: Coin Flip Game Worth IV
Post by Pietro K.C. on Sep 7^th, 2002, 8:13pm

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. :D

Let me know if there is something else to this problem that I didn't get.

Title: Re: NEW PROBLEM: Coin Flip Game Worth IV
Post by rugga on Sep 9^th, 2002, 2:07am

That's the right answer. But I agree about poker :)