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Each flip is 50/50 regardless of history: the coin doesn't remember (the gambler's fallacy in one sentence). But streaks are more common than intuition expects: in 100 flips, a run of 5 somewhere is near-certain and a run of 7 more likely than not: the difference between "this exact sequence from here" and "somewhere in a long session".
What The Coin Flip Streak Calculator Works Out
- Probability: the chance of the next n flips all landing heads, which is 0.5 to the power of n. Five in a row is 1 in 32, ten is 1 in 1,024 and twenty is roughly one in a million.
- Odds: the same figure written as odds against: 2 to the power of n, minus 1. Five heads is 31 to 1 against.
- Perspective: "this exact streak starting now" and "some streak somewhere in a long session" are different questions. A hundred flips almost always contain a run of five, and more often than not a run of seven. Conflating the two powers most bad statistical intuition.
- Flips in the whole session (advanced): the calculator works out exactly (not by approximation) how likely at least one run of your length is somewhere in that many flips. This is the figure to quote when someone presents a streak as evidence: ask how many chances it had to happen.
- Chance of heads on each flip (advanced): a biased coin, or any repeated event with a fixed probability: a 60% free-throw shooter, a 48.6% red on a European roulette wheel, a 52% stock-tipster. The streak maths is the same.
- Heads or tails (advanced): count a run of either face. From a standing start this doubles the chance for a fair coin (the first flip can be anything, the rest must match), and runs of either kind are what people notice at the table.
- Expected flips to first streak (advanced): the average number of flips until the first run of n heads, which for a fair coin is 2 to the power of (n + 1), minus 2: 62 flips for a run of five, 2,046 for a run of ten.
- Longest run you should expect (advanced): the longest run that is more likely than not to appear in the session, roughly log2 of the number of flips. Hand-faked "random" sequences betray themselves by having too few long runs; true randomness clumps.
- Expected heads in the session (advanced): flips x probability, with the typical spread of two standard deviations. In 100 fair flips anything from 40 to 60 heads is unremarkable.
- Streak table (advanced): run lengths 1 to 12 with the chance from a standing start and the chance somewhere in the session, side by side, so the gap between the two columns is visible.