In a run of n coin flips the longest streak of the same face lands near log₂n — about six or seven in a hundred flips — so a streak that looks rigged is usually the ordinary case.
There are many places in a long sequence for a streak to start, and the chance of at least one long streak appearing somewhere grows with the number of opportunities even though each individual flip stays at fifty-fifty.
It is a description of what randomness looks like, not a test of it. A genuinely biased coin also produces long runs, so a long streak on its own neither proves nor disproves anything — that needs a goodness-of-fit test over many flips. It also tempts people into the opposite error: believing a streak must end soon, when the next flip is always fifty-fifty.
At 100 the rule says 6.644 and the exact answer is 5.977 — 11% high. It is never within 5% anywhere in this range.
log₂(n) is the leading term and there are two more after it, worth about −0.67 between them. So the rule always predicts a longer streak than really turns up: about 6.6 in a hundred flips against a true 6.0. It is still the right answer to the question people are actually asking — a run of six is ordinary.
The rule against the exact answer, computed across the range. Inside the shaded band the shortcut is close enough to use; outside it, reach for the calculator.
Estimate with the rule, then check it against the calculator that models it properly.
Open Dice Roller & Coin Toss →In a run of n coin flips the longest streak of the same face lands near log₂n — about six or seven in a hundred flips — so a streak that looks rigged is usually the ordinary case. There are many places in a long sequence for a streak to start, and the chance of at least one long streak appearing somewhere grows with the number of opportunities even though each individual flip stays at fifty-fifty.