Coin Flip Probability Calculator
Compute the binomial probability of getting exactly, at least, or at most a chosen number of heads when you flip a coin a set number of times.
Binomial chance of getting precisely this many heads
- 1
Combinations C(n, k)
10! ÷ (6! × 4!) = 210Number of ways to choose which flips land heads. - 2
Heads factor p^k
0.5⁶ = 0.015625 - 3
Tails factor (1−p)^(n−k)
0.5⁴ = 0.0625 - 4
P(X = k) as percentage
(210 × 0.015625 × 0.0625) × 100 = 20.5078Multiplied by 100 to express as a percentage.
How does this calculator work?
For n coin flips with heads probability p, the chance of exactly k heads is C(n, k)·p^k·(1 − p)^(n − k). Sum terms for "at least" (i = k..n) or "at most" (i = 0..k). The expected number of heads is n·p with standard deviation sqrt(n·p·(1 − p)).
Formula
How this is calculated
Each coin flip is an independent Bernoulli trial with probability p of landing heads (p = 0.5 for a fair coin). The number of heads X in n flips follows a binomial distribution. The probability of exactly k heads is C(n, k) · p^k · (1 − p)^(n − k), where C(n, k) counts the number of ways to choose which k of the n flips are heads.
The cumulative results sum individual probabilities: P(X ≥ k) adds the mass for i = k up to n, and P(X ≤ k) adds the mass for i = 0 up to k. The distribution's mean (expected number of heads) is n·p, and its standard deviation is sqrt(n·p·(1 − p)), which measures the typical spread around that mean.
Inputs must satisfy 0 ≤ k ≤ n and 0 ≤ p ≤ 1; otherwise the result is undefined. To avoid overflow with large factorials, the calculator computes the binomial coefficient through log-gamma (sum of logarithms) and exponentiates at the end. The PMF plot shows the probability for every possible heads count from 0 to n.
Frequently asked questions
With n = 10, k = 5 and p = 0.5, P(X = 5) = C(10, 5)·0.5^10 ≈ 24.6%, the single most likely outcome for a fair coin.
Set p to the coin's true chance of heads. For example p = 0.6 means heads are 60% likely on each flip; the mean shifts to 0.6·n and the whole distribution leans toward more heads.
P(X = k) is the chance of precisely k heads, while P(X ≥ k) sums the chances of k, k+1, ... up to n heads. "At least" is always greater than or equal to "exactly".
Also known as
TG we-Calculate Editorial Team. (2026). Coin Flip Probability Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/coin-flip-probability
TG we-Calculate Editorial Team. "Coin Flip Probability Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/coin-flip-probability.
TG we-Calculate Editorial Team, "Coin Flip Probability Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/coin-flip-probability
@misc{wecalculate_coin_flip_probability, title = {Coin Flip Probability Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/coin-flip-probability}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?