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?