LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A fair coin is flipped until the first tails, and the payout is 2 raised to the number of flips. The expected payout of that game is infinite. The number of flips itself has a finite expectation of 2. Sample averages of the flip count settle near 2 as you repeat the game. Sample averages of the payout do not settle. They jump to a new level when a rare long streak arrives, which is why the lecture calls the story a failure of consistency rather than a pile of infinite cash. The usual limit theorems need a finite mean, and here the payout does not have one.
Why it matters
The law of large numbers says that as the number of trials grows, the sample average converges in probability to the population mean, when that mean exists as a finite number. The central limit theorem then describes the bell around that mean. People use both theorems every time they treat an average of simulated payoffs as “what the game is worth.” This game is the counterexample the lecture walks through, under the name of the St. Petersburg paradox. If you ask what you should pay to play, the expectation does not name a finite ticket price. A ticket price read off one chart of averages is an artifact of which rare streak showed up in that chart.
The idea in plain words
Stop at the first tails. If that happens on flip 1, the payout is 2. If it happens on flip 4, the payout is 2 to the fourth, which is 16. The tree does not end. After every heads you flip again, and the prize doubles in the exponent.
The flip count N is geometric: independent trials, success equal to tails. The probability that the first tails arrives on trial t is (1/2) to the t, which is the lecture’s (1/2) times (1/2) to the t minus 1. For a geometric count with success probability 1/2, the expected count is 1 divided by 1/2, which is 2. On average you expect tails in two flips. That expectation is finite, and the probability mass function is an ordinary formula. NumPy can draw the count directly. Repeated draws in the lecture came back as one flip, two flips, three flips, and so on.
The law of total expectation writes the game’s value as a sum over t of the payout given N equals t, times the probability that N equals t. Given N equals t, the payout is not random. It is 2 to the t. Multiply by the geometric probability, simplify, and each term equals 1. The sum of 1 from t equals 1 to infinity is infinite. So the expected payout is infinite. “We expect to earn infinity” is the sentence the lecture rejects. There is no finite mean to expect. The algebra produced an infinite sum of ones, which means the mean does not exist as a real number.
How to use it
Before you average simulated payoffs, ask whether the expectation you are estimating is finite. If you can sum the series and every term stays order 1, stop and do not trust the Monte Carlo mean. The geometric count is the control: its mean is 2, and the running average should approach 2. Use that control in the same notebook so a coding bug is not mistaken for the paradox.
What usually goes wrong
Saying the player should expect infinite wealth treats an infinite sum as a paycheck. The lecture’s point is the opposite. The expectation does not settle, so it does not tell you what you will earn.
Questions people ask
What is the expected payout?
Infinite. Each possible stopping time t contributes a term that simplifies to 1, and the sum of those terms diverges. The expected number of flips is a different object, and it equals 2.
What does a long simulation of the payout do?
It refuses to settle. The lecture’s reruns hovered in different bands, including about 12 to 14, about 10 to 12, and about 15 to 20. A rare long streak moves the average. There is no finite mean for it to approach.
Which theorem fails?
The large-sample law for averages, and the theorem that turns those averages into a bell, because both need a finite expectation. The sample average is not a stand-in for a population mean when that mean is infinite.