Coin Flip Probability Calculator
The probability of getting a given number of heads in a run of coin flips — exactly, at least, at most — for a fair or a biased coin.
Each flip is independent, so a run of n flips is a binomial experiment.
How the coin flip probability calculator works
Each flip is independent, so a run of n flips is a binomial experiment. The chance of exactly k heads is C(n, k) p^k (1 − p)^(n − k), where p is the chance of heads on one flip. The calculator sums that over the relevant range for the "at least" and "at most" answers.
The result that surprises people: with 10 flips of a fair coin, exactly 5 heads is only about 24.6% likely. "Half heads" is the single most likely outcome, but it is far from the expected one.
Formula: P(X = k) = C(n, k) p^k (1 − p)^(n − k)
Worked examples
| Inputs | P(exactly k heads) | Note |
|---|---|---|
| 5 heads in 10 flips | 24.6094% | about 24.6% |
| 10 heads in 10 flips | 0.0977% | 1 in 1,024 |
| 3 heads in 5 flips | 31.25% | 31.25% |
FAQFrequently asked questions
What is the chance of 5 heads in 10 flips?
About 24.6% — the most likely single outcome, but still far from certain. Getting between 4 and 6 heads is about 66%.
What is the chance of 10 heads in a row?
1 in 1,024, or about 0.098%.
Does a coin remember previous flips?
No. Each flip is independent, so after nine heads the tenth is still 50/50. Believing otherwize is the gambler's fallacy.
Are real coins fair?
Very nearly. A spun coin can be biased by its edge design, and a caught flip has a slight bias toward the face it started on — around 51%.
Why is exactly half so unlikely?
Because the probability is spread across every possible count. With more flips the *proportion* concentrates near half but the chance of hitting exactly half falls.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.