Fair game paradox

Question

A fair coin is flipped repeatedly until the first heads appears. The payout is $2^n$ dollars if the first heads occurs on the $n$th toss. Determine the fair value of this game. If the expected value is infinite, output $-1$.

Accepted Answer

Let $N$ be the number of coin flips until the first heads appears. Since the coin is fair, $N$ follows a geometric distribution with success probability $1/2$ : $P(N = n) = (1/2)^n$ for $n = 1, 2, 3, \dots$ (the first $n-1$ flips are tails and the $n$ th flip is heads). The payout when $N = n$ is $2^n$ dollars. The expected value of the payout is

E[\text{payout}] = \sum_{n=1}^{\infty} 2^n \cdot P(N = n) = \sum_{n=1}^{\infty} 2^n \cdot \left(\frac{1}{2}\right)^n = \sum_{n=1}^{\infty} 1 = \infty.

Since the expected value is infinite, the game does not have a finite fair value. Per the problem instructions, output $-1$ .

Fair game paradox

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