# A variation on an old paradox

Probability Level 3

The infamous "St. Petersburg Paradox", analyzed by the Bernoulli family, is as follows:

I flip a coin. If it first comes up heads on the first toss I pay you $2, on the second I pay you$4, on the third $8, and so on. How much is a fair price for me to charge you to play? If we calculate the mathematical expectation, your expectation is $(\2 * 1/2) + (\4 * 1/4) + (\8 * 1/8)... = \1 + \1 + \1....$ out to infinity, or in other words, infinite. So no amount you can pay me is fair. Now let's modify the situation as follows. If the first head appears on the first or second flip I pay you$2, on the third or fourth flip I pay you $4, on the 5th or 6th flip I pay you$8, and so forth. What is a fair price for me to charge you, so that the price is exactly equal to your mathematical expectation? Give answer in dollars.

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