Coin Flip Probability Calculator

A coin flip is the simplest probability experiment there is, and the counting behind “exactly k heads” and “at least one” still catches most people out. The formulas below are the whole of it.

Single flip: P(Heads) = 0.5 (50%) for a fair coin.

Multiple independent flips: P(All Heads in n flips) = 0.5ⁿ

P(At least one Heads in n flips) = 1 - 0.5ⁿ

Exact count using Binomial Distribution: P(exactly k heads in n flips) = C(n,k) × 0.5ⁿ

Where C(n,k) = n! ÷ (k! × (n-k)!), the number of ways to choose k items from n.

Worked examples:

Q: What’s the chance of getting exactly 3 heads in 5 flips? C(5,3) = 10 P = 10 × 0.5⁵ = 10 × 0.03125 = 0.3125 (31.25%)

Q: What’s the chance of flipping 5 heads in a row? P = 0.5⁵ = 0.03125 (3.125%), about 1 in 32

Q: If you flip 10 times, what’s the chance of getting at least one heads? P = 1 - 0.5¹⁰ = 1 - 0.000977 = 99.9%

Notice that the second and third questions are the same calculation from opposite ends. “All heads” is the exact case k = n, and “at least one head” is 1 minus the exact case k = 0. Any “at least” question is easier read as one minus its complement, which is why the calculator gives you both.

The Gambler’s Fallacy Each flip is independent. If you have flipped 9 heads in a row, the 10th flip is still exactly 50/50. The coin has no memory. Past results do not influence future flips, and no run is ever “due” to end.

The confusing part is that a long run of heads really does become rarer the longer it goes, and that is not a contradiction. Before you start, 10 heads in a row is 1 in 1,024. After nine heads have already landed, the tenth is a fresh coin: 1 in 2. The improbability was spent on the flips you already watched.

Nothing here is flipped. The page computes the odds from the binomial formula, so the same inputs always give the same answer.
To watch actual flips land and see the streaks come out, use the flip a coin page, which simulates up to 1,000 throws and reports how far the result strayed from what this page predicts.

Expected number of heads in n flips: Expected Heads = n × 0.5, with standard deviation √n / 2. In 100 flips you expect 50 heads, give or take about 5. Anything from roughly 40 to 60 is ordinary, and the result panel reports how many standard deviations your own target sits from the mean.