Ergodicity
The time-average of one person's path is not the ensemble-average across many people.
A subtle but crucial distinction. If you play a game where you have a 50% chance of doubling your money and a 50% chance of losing 60%, the ensemble-average across many players is positive. The time-average for any single player is ruin: in the long run, you go broke.
The expected-value calculation gives the wrong answer when outcomes include ruin. Because you only get one trajectory, what matters is the time average, not the ensemble average. The ensemble doesn't help if your path leads to zero.
For operators, the lesson is brutal: any bet that includes a non-trivial chance of ruin has wrong expected value, no matter how attractive the math looks on a single sample. You don't get to average across many lives.
Examples in the wild
Going "all in" on a single bet can be a positive-expected-value move and still wrong, because if you lose, you don't get another shot. The time-path matters more than the ensemble math.
Leveraged returns that average well across all possible markets fail when applied to a single path that hits ruin. Most leveraged ETFs lose money over time despite positive ensemble math.
Driving without a seatbelt has positive expected value (nothing happens 99.99% of trips). The time-path matters. One trip can end the game.
Ergodicity is one of the mental models we apply through real cases inside the Pareto MBA — a part-time program for professionals who want to think clearly about business.