Expected value stops being good advice when the loss is permanent

What expected value actually is

Expected value is the average result of a gamble if it were repeated an unlimited number of times: each possible outcome multiplied by its probability, all added together. It is the correct guide in a large class of situations, and it deserves its reputation, because reasoning that ignores it produces obvious errors that cost real money.

But the definition contains a condition that is easy to miss. The average is over repetitions. It describes what happens to a long sequence of similar bets, and it says remarkably little about any single one. When a decision will be faced many times, and no individual outcome can prevent the next attempt, expected value is close to the whole answer. Change either of those conditions and it stops being sufficient.

The absorbing state

Some losses end the sequence. A business that runs out of cash does not get to make the next decision; a person whose reserves are gone cannot wait for the average to assert itself; an irreversible physical outcome removes every option that followed it. In the formal language these are absorbing states, and once entered they cannot be left.

This changes the arithmetic in a way that is not a matter of temperament. A strategy with a positive expected value per attempt can still converge on ruin with near certainty when repeated, if each attempt carries some chance of losing everything, because the chances of survival multiply and a product of numbers below one falls towards zero. The average across parallel worlds stays attractive while the typical path goes broke. This is not a paradox; it is what happens when averaging over outcomes is confused with averaging over time.

Why the sizing of the bet does most of the work

It follows that how much you stake matters at least as much as whether the odds are favourable. A favourable bet at a small fraction of your resources can be repeated indefinitely and will do what the average promises. The same bet at a large fraction of your resources is a different proposition entirely, even though the odds have not moved at all.

There is a well-developed body of mathematics on how much to stake given a known edge, and its details are contested and depend heavily on assumptions that rarely hold outside a casino. The general shape of its advice is not contested: bet a fraction rather than a sum, and a smaller fraction the less certain you are about your edge. Overestimating your advantage is punished far more severely than underestimating it, which is a strong argument for caution when the edge is estimated rather than known.

Insurance stops looking irrational

Buying insurance has a negative expected value by construction. The premiums collected must exceed the claims paid, or the insurer would not exist, so on average the buyer loses. Judged purely by expected value, insurance is a mistake, which should be a clue that expected value is not the whole framework.

What insurance buys is the removal of the tail — the outcomes large enough to end the sequence. Paying a small, certain, survivable amount to eliminate a small chance of an unsurvivable one is a sensible trade for anyone who cares about continuing to play, and the negative average is the price of that. It follows that insurance makes most sense precisely where the potential loss is large relative to your resources, and least sense where the loss would be an annoyance. Extended warranties on inexpensive goods fail that test almost by definition.

Reading a decision for this property

The question to ask is not only whether the odds are good but whether the bad outcome removes your ability to continue. If it does, the size of the stake and the availability of a recovery path matter more than the expected value, and a favourable average is not a defence.

If it does not — if the worst case is a bad quarter, a wasted weekend, an embarrassing month — then the sequence continues and the average is a good guide. In those cases the more common error is the opposite one: refusing repeated favourable bets because each individual loss is unpleasant, which forgoes a real gain in exchange for avoiding a feeling. Both errors come from the same source, which is not asking whether this decision is one of many or one of one.