A coin is flipped until the first tail; you win $2^n where n = heads before that tail. Fair price?

How to answer

Infinite — the St. Petersburg paradox. P(n heads then a tail) = (1/2)^{n+1}, payout 2^n, so E = Σ_{n≥0} 2^n·(1/2)^{n+1} = Σ 1/2 = ∞. The expectation diverges, so no finite fair price exists under pure EV; sensible pricing needs utility/risk adjustment or a finite bankroll cap (log utility yields a small finite value).

Key idea: Trying to sum to a finite number, or quoting a price without noting the expectation literally diverges and why utility/bankroll bounds it.