See Example 4.21 on the St. Petersburg paradox. Modify the game so that you only receive $210 = $1024 if any number greater than or equal to 10 tosses are required to obtain the first tail. Find the expected value of this game.

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 €œparadox€ is that most people would not pay very much to play this game. And yet the expected payout is infinite.

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EX]Σ2, >1= +00. Σ1- 2η n=1 n=1