Reading Streaks Without Fooling Yourself

People are extraordinarily bad at recognising randomness. Asked to write down a 'random' sequence of colours, almost everyone alternates too much and produces far fewer long runs than chance actually delivers. Real randomness is clumpy, and the clumps are the part that feels wrong.

Start with the arithmetic. On a six-colour die the chance of matching the previous face is 1/6. So a run of exactly five identical faces starting at a given point has probability (1/6)^4 x (5/6), about 0.08%. That sounds vanishingly rare until you remember you are scanning a whole sequence: over a thousand rolls you get roughly a thousand chances for such a run to start, and you should expect to see one.

A useful rule of thumb: in n rolls of a six-sided die, the longest run you should expect is about log(n) divided by log(6), plus a bit. For 100 rolls that is roughly 3; for 1,000 rolls roughly 4; for 10,000 rolls roughly 5. Seeing a run of four in an evening of play is not evidence of anything.

What does real bias look like? Not streaks — frequencies. If one colour is genuinely favoured, its share of the total drifts away from 16.67% and stays there. The noise in that share shrinks like one over the square root of the sample size, which is the number that matters when you decide how much data you need.

Concretely: with 100 rolls, a fair colour will land anywhere between about 9% and 24% of the time without anything being wrong. With 1,000 rolls the honest band narrows to roughly 14% to 19%. With 10,000 rolls, roughly 16% to 17.4%. If you want to accuse a roller of bias, you need thousands of rolls, not a bad night.

A cheap self-check anyone can run: roll 25 dice forty times, which gives you a thousand faces, and tally the six colours from the stats panel. Every colour should land somewhere near 167. One colour at 250 is a story. One colour at 190 is a Tuesday.

The last trap is stopping rules. If you keep rolling until you see the pattern you wanted and then stop, you will always find it, and the statistics you compute afterwards are meaningless. Decide how many rolls you will count before you start counting.