Winning streak probability calculator
Long streaks are far more likely than intuition suggests. In a hundred coin flips, a run of six heads is more likely than not.
%
Chance of the streak occurring
77.336%
- Chance of it starting at a given trial
- 1.5625 %
- Expected number of such streaks
- 1.4844
- Expected longest streak
- 5.64
- One in how many, for a single start
- 64.0
- Trials for an even chance
- 44
Show the working
- Chance of a win = 50
- Streak length = 6
- Number of trials = 100
- p = 50 / 100 = 0.5
- Chance of it starting at a given trial = 0.5^6 * 100 = 1.5625
- Expected number of such streaks = max(100 - 6 + 1, 0) * 0.5^6 = 1.484 38
- Chance of the streak occurring = (1 - exp(-1.484 38)) * 100 = 77.3356
- Expected longest streak = 0.5 < 1 ? log(100 * (1 - 0.5), 1/0.5) : 100 = 5.643 86
- One in how many, for a single start = 1.5625 > 0 ? 100/1.5625 : 0 = 64
- Trials for an even chance = 0.5 > 0 ? ln(2) / 0.5^6 : 0 = 44.3614
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- expected streaks = (trials - streak + 1) x p^streak
- P(at least one) is approximately 1 - exp(-expected streaks)
- expected longest run is about log(n(1-p)) / log(1/p)
The Poisson approximation used here is very good for rare streaks and slightly optimistic for common ones.
A streak in past results tells you nothing about the next trial if the trials are genuinely independent.
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