Independent trials: why the last spin tells you nothing

An independent trial has no memory, which is why a run of one colour changes nothing about the next spin. The intuition that says otherwise is common enough to have a name, and naming it precisely makes it easier to catch in yourself.

What independence means

Two events are independent when knowing the outcome of one does not change the probability of the other. A roulette wheel, a coin and a shuffled-every-hand deck all satisfy this. The physical apparatus carries no record of what it did last time, so the next outcome is drawn from exactly the same distribution as the last one.

The two mirrored errors

The first error says an outcome is due after a run against it. The second says a run predicts more of the same, because the wheel is somehow favouring it. Both treat past outcomes as information about the next one. Under independence, neither can be true, and they cannot both be true at once in any case.

Where the intuition comes from

Long-run proportions really do settle down, and that is a genuine result. But they settle by dilution rather than by correction. Later trials do not compensate for an early imbalance; there are simply so many of them that the early imbalance becomes a small share of the total. Nothing pushes back toward the average.

The one real exception

Dependence appears when the apparatus genuinely changes between trials, as it does in a game dealt from a shoe that is not reshuffled. There, earlier cards really have left the deck, so the remaining composition really has changed. That is a different situation from a wheel, and it is worth keeping the two apart rather than generalising from one to the other.