See Example 4.21 on the St. Petersburg paradox. Modify the game so that you only receive $210
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Data from Example 4.21:
I offer you the following game. Flip a coin until heads appears. If it takes n tosses, I will pay you $2n. Thus if heads comes up on the first toss, I pay you $2. If it first comes up on the 10th toss, I pay you $1024. How much would you pay to play this game? Would you pay $5, $50, $500? Let X be the payout. Your expected payout is
The expected value is infinite. The expectation does not exist. This problem, discovered by the eighteenth-century Swiss mathematician Daniel Bernoulli, is the St. Petersburg paradox. The paradox is that most people would not pay very much to play this game. And yet the expected payout is infinite.
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