A fair coin is tossed until heads appears for the first time. If that happens on toss number $n$, the game pays $$2^n$. What is the fair (expected) value of playing? Give $-1$ as your answer if it is infinite.

Fair price of the doubling-payout coin game

Question

A fair coin is tossed until heads appears for the first time. If that happens on toss number $n$ , the game pays $\text{\char36}2^n$ . What is the fair (expected) value of playing? Give $-1$ as your answer if it is infinite.

Short answer

Infinite (answer $-1$ ). The first head lands on toss $n$ with probability $2^{-n}$ and pays $2^n$ , so every $n$ contributes exactly $1$ to the expectation and the series $\sum_{n\ge1}1$ diverges. This is the St. Petersburg paradox.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Sample Spaces, Events, and the Probability Axioms

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free