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If the expected value of a bet is positive, should you do it? In other words, if you will make money on average, should you play?

How about this: A game costs 1 million dollars to play. The amount of money you get will be determined by flipping a fair coin. Your winnings depend on how many heads in a row you get. If you get a tail the game is over.
Here is the payout:

0 heads - 1$
1 head - 2$
2 heads - 4$
3 heads - 8$
4 heads - 16$
5 heads - 32$

If you make h heads in a row, your return is 2h dollars.

Expected value is the sum of (the probability of an outcome * it's value) for all outcomes.

That is 1/2*1 + 1/4*2 + 1/8*4 + 1/16*8 ...
= 1/2 + 1/2 + 1/2 ...

So your expected value is infinite. However, your chances of coming out ahead in this game are less than one in a million. The thing is, even though you have a low chance of coming out ahead, if you do manage to make money, you'll make so much that it'll balance out all those times you lose.

This still works if I set the first million payouts to zero - if I set it so you start getting payouts once you make 1,000,000 heads in a row. Expectation is still infinite, but in this case your chances of making any money are 1 in 21,000,000. Anybody who wants to play, send me a message...

I think this shows that judging the value of a game just based on expected value can be wrong. Perhaps this can be applied to Pascal's wager.

Say that when you die there's a 50% chance of going to heaven. If someone offers you a deal: if you spend 1 million years getting tortured, you'll have a 50.001% chance of getting into heaven. You pay a finite price for an infinite gain (the .001% extra chance to get the infinite value of heaven has an infinite value...). Yet, I think most people wouldn't do it.

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