Coin Flip Probability Calculator
Enter the number of flips, the number of heads you want to hit, and whether you need exactly that many, at least that many, or at most that many. The calculator applies the binomial formula and returns the probability as a decimal, a percentage, and an odds ratio. You can also set a custom coin bias if the coin is not fair. The distribution chart shows the full probability spread across every possible outcome.
Formula
Worked example
You flip a fair coin 10 times and want exactly 6 heads. C(10,6) = 210, p^6 = 0.5^6 = 0.015625, (1-p)^4 = 0.5^4 = 0.0625. Multiply: 210 x 0.015625 x 0.0625 = 0.2051, or about 20.5%. Expected heads = 10 x 0.5 = 5; standard deviation = sqrt(10 x 0.5 x 0.5) = 1.58.
How the binomial probability formula works
Every coin flip is an independent Bernoulli trial: it lands heads with probability p and tails with probability 1-p. When you repeat that trial n times, the number of heads follows a binomial distribution. The formula P(X = k) = C(n,k) x p^k x (1-p)^(n-k) has three parts. The combination C(n,k) counts how many different orderings of k heads and n-k tails exist. The term p^k is the probability of those k heads occurring. The term (1-p)^(n-k) is the probability that the remaining n-k flips all land tails. Multiplying them gives the total probability of seeing exactly k heads in any order.
Exactly, at least, and at most
The outcome type controls which region of the distribution you care about. "Exactly k" returns the PMF value at k. "At least k" returns the sum of P(X = i) for i from k to n, which equals 1 minus the cumulative probability up to k-1. "At most k" returns the cumulative probability P(X <= k), the sum from i = 0 to k. For small n (under about 50) all three modes compute exactly. For large n the probabilities of individual outcomes become very small; the expected value and standard deviation help you reason about where the bulk of outcomes will fall.
Expected value, standard deviation and the normal approximation
The expected number of heads is mu = n x p. For a fair coin flipped 10 times, mu = 5. The standard deviation sigma = sqrt(n x p x (1-p)) measures how spread out the results typically are. For the same 10-flip fair-coin example, sigma is about 1.58, meaning most outcomes fall between 3 and 7 heads. When n is large and p is not too close to 0 or 1, the binomial distribution is well approximated by a normal distribution with the same mean and variance. A common rule of thumb is that the approximation is reasonable when n x p >= 5 and n x (1-p) >= 5.
Biased coins and real-world applications
A real coin is very close to fair but not perfectly so. Research suggests a typical coin lands with the starting-face-up side slightly more often (roughly 51% rather than 50%), though the difference is tiny in practice. The bias field in this calculator lets you explore asymmetric probabilities: a value above 0.5 represents a heads-heavy coin, and a value below 0.5 represents a tails-heavy one. Beyond coin flipping, the same binomial formula applies to any sequence of independent yes/no trials: free throws in basketball, quality-control defect checks, clinical trial response rates, A/B test conversions, and genetics (dominant vs. recessive alleles in a cross).
Common coin-flip probability benchmarks
| Scenario | Probability | Approx. odds |
|---|---|---|
| Heads on 1 flip | 50% | 1 in 2 |
| 2 heads in 2 flips | 25% | 1 in 4 |
| 3 heads in 3 flips | 12.5% | 1 in 8 |
| Exactly 5 heads in 10 flips | 24.6% | 1 in 4.1 |
| At least 7 heads in 10 flips | 17.2% | 1 in 5.8 |
| 10 heads in 10 flips | 0.098% | 1 in 1,024 |
| Exactly 50 heads in 100 flips | 7.96% | 1 in 12.6 |
| 20 heads in 20 flips | ~0.0001% | 1 in ~1 million |
Probabilities for getting exactly k heads in n flips of a fair coin (p = 0.5). Fractions shown for familiar benchmarks.
Frequently asked questions
What is the probability of getting exactly 5 heads in 10 flips?
For a fair coin, P(X = 5) = C(10,5) x 0.5^5 x 0.5^5 = 252 x (1/1024) = 252/1024, which is approximately 24.6%. Despite being the single most likely individual outcome, it still happens less than 1 in 4 times, because there are 11 possible outcomes (0 through 10 heads).
How does the "at least" mode work?
"At least k heads" means k, k+1, k+2, ..., or n heads. The probability is the sum of P(X = i) for each i from k to n, which is the same as 1 minus P(X <= k-1). For example, "at least 7 heads in 10 fair-coin flips" = 1 - P(X <= 6) = about 17.2%.
What does standard deviation mean in coin flipping?
The standard deviation sigma = sqrt(n x p x (1-p)) tells you how much the number of heads typically varies from the expected value. For 100 fair-coin flips, sigma = 5, so you would expect most runs to land between about 45 and 55 heads. About 68% of runs fall within one sigma of the mean, and about 95% fall within two sigma.
Can this calculator handle a biased or unfair coin?
Yes. Set the "probability of heads" input to any value between 0 and 1. A value of 0.5 is a fair coin. A value of 0.6 means the coin lands heads 60% of the time, and the binomial formula adjusts p and (1-p) throughout.
Is there a quick way to see every possible outcome at once?
Yes: the distribution chart below the result shows P(X = k) for every value of k from 0 to n (up to n = 200). The bars illustrate the bell-shaped spread around the expected value, and you can visually see how rare the tails are. Changing n or p updates the chart instantly.
Why does the probability of all heads decrease so quickly as n increases?
For a fair coin, P(all heads in n flips) = (0.5)^n. That halves with every extra flip: 50% for 1 flip, 25% for 2, 12.5% for 3, and so on. At 10 flips it is about 0.098%, and at 20 flips it is roughly 1 in a million. Exponential decay means small probabilities become astronomically small very quickly.