Coin Flip Calculator
The Simplest Random Experiment, Solved Exactly
A coin flip is the textbook example of a random event for a reason: it has exactly two outcomes, and (for a fair coin) they're equally likely. But questions like "what's the probability of getting at least 7 heads in 10 flips" aren't as obvious as they sound — they require the binomial distribution, not simple multiplication, because there are many different orderings of heads and tails that produce the same total count.
The Formula
n is the number of flips, k is the number of heads, and p is the probability of heads on a single flip (0.5 for a fair coin). C(n, k) counts how many different sequences of flips produce exactly k heads.
The Full Distribution for 10 Fair Flips
| Heads (k) | Probability |
|---|---|
| 0 | 0.10% |
| 1 | 0.98% |
| 2 | 4.39% |
| 3 | 11.72% |
| 4 | 20.51% |
| 5 | 24.61% |
| 6 | 20.51% |
| 7 | 11.72% |
| 8 | 4.39% |
| 9 | 0.98% |
| 10 | 0.10% |
Exactly 5 heads out of 10 — the most "expected" outcome — only happens about a quarter of the time. The probability of at least 7 heads is 17.19%, higher than intuition usually suggests.
Where This Matters
- Teaching probability — coin flips are the standard entry point for understanding the binomial distribution.
- Game design — balancing mechanics that rely on repeated 50/50 (or biased) chance events.
- Testing for bias — if a coin comes up heads far more than the binomial distribution predicts, that's evidence it isn't fair.
- Simulation sanity checks — verifying that a random number generator produces results consistent with theoretical probability.
How to Use This Calculator
- Choose Exactly, At Least, or At Most k Heads from the dropdown.
- Enter the number of flips.
- Enter the number of heads (k) you're interested in.
- Optionally set a probability of heads other than the default 50% for a biased coin.
- Select Calculate to get the probability.
Related Calculations
For probability problems that aren't restricted to two outcomes, see the Binomial Distribution Calculator. To work with a die instead of a coin, use the Dice Roll Calculator.