Coin Flip Probability Calculator
Get the exact probability of any heads count in a run of fair coin flips. Enter the number of flips and a target number of heads, then choose exactly, at least, or at most — the tool returns the percentage, the 1-in-X odds, and the decimal probability.
Example: with Number of flips (n) 10 · Number of heads (k) 5 · Question Exactly k heads → Probability: 24.61%.
- OddsAbout 1 in 4.1
- Decimal probability0.246094
Computed by the calculator below using its default values. Change any input to see your own numbers.
Exact binomial math with p = 1/2: P(exactly k) = C(n, k) / 2ⁿ. No normal approximation is used — sums are computed term by term.
The binomial math behind coin flips
Each fair flip is heads with probability 1/2, and every specific sequence of n flips is equally likely: there are 2ⁿ of them. Getting exactly k heads can happen in C(n, k) different orders, so P(exactly k) = C(n, k) / 2ⁿ. For 5 heads in 10 flips that is 252 / 1,024 = 24.61% — the single most likely count, yet it still misses three times out of four.
"At least" and "at most" questions just add the exact probabilities term by term. That is why at least 8 heads in 10 flips is (45 + 10 + 1) / 1,024 = 5.47%: rare, but not remarkable. Runs of coincidence look impressive mostly because we forget how many chances they had to happen.
How it’s calculated
P(exactly k heads in n flips) = C(n, k) / 2ⁿ. Cumulative modes sum those exact terms: P(at least k) = Σ from j = k to n, P(at most k) = Σ from j = 0 to k. Binomial coefficients are built by the recurrence C(n, j) = C(n, j−1) × (n − j + 1) / j, so results are exact (no normal approximation) up to n = 1,000.
Assumes a fair coin (p = 0.5) and independent flips — real coins are close to fair, but a coin caught mid-air is not measurably biased by starting side in normal use.
Exactly k heads in 10 flips
| Heads (k) | Ways C(10, k) | Probability |
|---|---|---|
| 0 | 1 | 0.10% |
| 1 | 10 | 0.98% |
| 2 | 45 | 4.39% |
| 3 | 120 | 11.72% |
| 4 | 210 | 20.51% |
| 5 | 252 | 24.61% |
Computed with C(10, k) / 1,024; the distribution is symmetric, so k = 6 matches k = 4, and so on.
Common mistakes
- Expecting exactly half heads to be likely — it is only the most likely single count (24.61% for 10 flips), not the majority outcome.
- The gambler’s fallacy: after 5 heads in a row, the next flip is still 50/50. Coins have no memory.
- Mixing up "exactly" with "at least" — at least 5 heads in 10 flips is 62.30%, not 24.61%.
- Multiplying 0.5 by n instead of raising it to the power n: five heads in a row is 0.5⁵ = 3.13%, not 2.5.
Frequently asked questions
What is the coin flip probability formula?
P(exactly k heads in n flips) = C(n, k) / 2ⁿ, where C(n, k) = n! / (k!(n − k)!) counts the orders the heads can arrive in. For at-least or at-most questions, add up the exact terms.
What are the odds of 10 heads in a row?
0.5¹⁰ = 1 / 1,024, about 0.098% or 1 in 1,024. Each additional head halves the probability.
If I flip 6 heads in a row, is tails due next?
No — that is the gambler’s fallacy. Each flip is independent, so the next one is 50/50 regardless of the streak. The streak was unlikely; the next flip is not.
What is the chance of exactly 50 heads in 100 flips?
C(100, 50) / 2¹⁰⁰ ≈ 7.96%. Even though 50 is the most likely count, the probability spreads across many nearby counts, so any single exact value is fairly rare.