Probability

Lesson 10 of 14

Infinite Expectations

A coin game with an infinite expected payout, partial sums as a walking dot, why the harmonic series diverges, and what breaks when E[X] does not exist.

A game with an infinite price

A casino offers you this game. A fair coin is flipped until it lands tails. If the first tails comes on flip kk, you win 2k2^k dollars: $2 if tails comes right away, $4 after one head, $8 after two heads, and so on. How much would you pay to play once?

Let XX be the payout. It equals 2k2^k with probability 2−k2^{-k}, so every outcome adds exactly one dollar to the expectation:

E[X]=∑k=1∞2−k⋅2k=1+1+1+⋯=∞E[X] = \sum_{k=1}^{\infty} 2^{-k} \cdot 2^k = 1 + 1 + 1 + \cdots = \infty

If the fair price of a bet is its expected payout, a price of $1,000 is a bargain, and so is any other finite price. Almost nobody would pay it. Half the time you win $2, and three quarters of the time you win $4 or less. To get $1,000 back in one game you need 2k≥10002^k \ge 1000, so k≥10k \ge 10: the first nine flips must all be heads. That has probability 2−9=1/5122^{-9} = 1/512, about 0.2%.

This is the St. Petersburg paradox, posed by Nicolaus Bernoulli in 1713 and named after Daniel Bernoulli's 1738 paper for the St. Petersburg Academy. The rest of this lesson asks what an infinite sum means, and when E[X]E[X] stops being a number you can use.