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I was discussing the the St. Petersburg paradox and the following question came up:

Suppose the game doesn't end within nine rounds, then the player directly receives $2^{10}$ dollars , while terminating the game at that moment. What would be the expected value in this example?

My approach:

$$E = [ 2\cdot\left(\frac{1}{2}\right)^1+...+2^9\cdot\left(\frac{1}{2}\right)^9 ] +2^{10}\cdot\left(\frac{1}{2}\right)^9$$ where the last term is due to the fact that we get $2^{10}$ dollars if the game is not finished after nine rounds which means we got nine times tail for which the probability should be $\left(\frac{1}{2}\right)^9$.
So in summary we get $E = 11$.

Am I right?

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    $\begingroup$ In a word, yes. $\endgroup$
    – David K
    Commented Nov 27, 2023 at 14:33
  • $\begingroup$ thank you! :-)) $\endgroup$
    – Blue2001
    Commented Nov 27, 2023 at 14:36

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