How coin flip probability is calculated
Coin flips follow the binomial distribution.
P(k) = C(n, k) × p^k × (1 − p)^(n − k)
- C(n, k) counts the ways to arrange k heads among n flips: n! ÷ (k! × (n − k)!).
- Multiply by the chance of heads to the power k and tails to the power n − k. For a fair coin that is 0.5ⁿ.
- For at least or at most, add up the chances for every count in that range.
- The sums are done with logarithms, so even 10,000 flips stay accurate without huge factorials.
Worked example — Exactly 5 heads in 10 fair flips: C(10, 5) = 252, and 252 × 0.5¹⁰ = 0.2461, or 24.61%, about 1 in 4.06.
Common coin flip questions
| Flips | Heads | Type | Probability | Odds |
|---|---|---|---|---|
| 2 | 2 | Exactly | 25% | 1 in 4 |
| 3 | 1 | At least | 87.5% | 1 in 1.14 |
| 10 | 5 | Exactly | 24.6094% | 1 in 4.06 |
| 10 | 10 | Exactly | 0.0977% | 1 in 1,024 |
| 20 | 15 | At least | 2.0695% | 1 in 48.32 |
| 100 | 60 | At least | 2.8444% | 1 in 35.16 |
Swipe the table sideways to see the rest.
Fair coin, 50% heads per flip.
Things to consider
- The single most likely result is half heads, but it gets less likely as flips grow.
- At least 1 head in n flips is 1 minus the chance of all tails.
- Past results do not change the next flip. A coin has no memory.
- Use the chance setting for weighted coins or any yes or no event with a fixed chance.
Frequently asked questions
What is the chance of 5 heads in 10 flips?
Exactly 5 heads in 10 fair flips is 24.61%, about 1 in 4.
What are the odds of 10 heads in a row?
0.5¹⁰ = 1 in 1,024, about 0.098%.
Does it work for a biased coin?
Yes. Change the chance of heads per flip, for example to 60%.
How many flips can it handle?
Up to 10,000 flips. It works in log space, so large numbers do not overflow.
After 5 tails, is heads more likely?
No. Each flip is independent, so a fair coin is still 50% heads. That belief is called the gambler's fallacy.
Results are estimates for general information only, not financial, medical or professional advice.