Paying the square of the number of heads

Topic: Functions of random variables and limit theorems · more questions on this topic
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Ten fair coins are tossed, and the payout is the square of the number of heads. What stake makes the game fair, in the sense that the expected profit is zero?

Worked answer (click to show)

Let X\sim\operatorname{Bin}(10,1/2). The payout is X^2, and

\mathbb{E} X^2=\operatorname{Var}(X)+(\mathbb{E} X)^2=10\cdot\tfrac12\cdot\tfrac12+5^2=2.5+25=27.5.

A stake of 27.5 makes the expected profit zero. Any smaller stake has positive expected profit for the player.