Probability · Dice & games
Coin Flip Probability Calculator
The odds of getting a given number of heads in a run of flips — exactly, at least, or at most — for a fair or a biased coin, with the whole distribution drawn out. There's also a streaks tab, because a run of heads is far more ordinary than it feels, and a separate flip is never “due” for tails.
How likely is at least one run of heads in a row?
Result
In plain English
Each flip is independent and has no memory, so a sequence of flips follows the binomial distribution: the number of heads piles up around half of your flips, with a predictable spread either side. Two facts about that spread routinely fool people — and both are on display here.
- exactly / at least / at most k
- The chance of precisely k heads, k or more, or k or fewer, in N flips. Even the single most likely count (half of N) is usually well under 50%.
- expected heads
- N × the chance of heads — the average count over many repeats, not a guarantee for one run.
- the two kinds of “evening out”
- The proportion of heads homes in on 50% as N grows (the law of large numbers). But the count of heads drifts further from exactly half, in absolute terms, as N grows. Both are true at once.
- streak / run
- A stretch of the same result in a row. Runs are a normal feature of randomness — in just 10 fair flips, three heads in a row is more likely than not.
- independence (no “due”)
- After any run of heads, the next flip is still 50/50. The coin isn't keeping score; expecting a correction is the gambler's fallacy.
Frequently asked
After five heads in a row, is tails more likely?
No — that's the gambler's fallacy. Each flip is independent and the coin has no memory, so tails stays at 50%. The feeling that a result is “due” to balance things out is exactly the trap: past flips can't reach forward and change the next one.
How likely is a streak of heads?
Much likelier than most people guess. In just 10 fair flips, a run of three or more heads happens more often than not. Long runs are an ordinary feature of randomness, not a sign the coin is rigged — which is why genuinely random data looks “too streaky” to our pattern-seeking eyes. Try the Streaks tab.
What's the probability of exactly k heads in N flips?
It's the binomial formula: C(N,k) · pᵏ · (1−p)^(N−k). For a fair coin, exactly 5 heads in 10 flips is about 24.6% — the single most likely outcome, yet still less than 1 in 4. “Half of them” is the most common result, not a typical guarantee.
What is the probability of at least one head in N flips?
One minus the probability of no heads at all: 1 − (1 − p)ⁿ. For a fair coin that is 1 − 0.5ⁿ — about 87.5% in 3 flips and 99.9% in 10. It climbs fast, which is why “at least once” events feel almost certain over many tries even when each single try is unlikely — the same maths behind why rare events do happen to someone, somewhere, all the time.