Coin Flip Probability Calculator
Find the probability of getting a given number of heads in a set of coin flips, for fair or biased coins.
Mode
Chance of exactly k heads
- 1
P(head)^k
0.5^5 = 0.03125 - 2
P(tail)^(n−k)
0.5^(10 − 5) = 0.03125 - 3
Binomial coefficient C(n, k)
10! ÷ (5! × 5!) = 252Number of ways to choose k heads from n flips. - 4
P(X = k)
252 × 0.03125 × 0.03125 = 24.6094%
How does this calculator work?
The chance of k heads in n coin flips is C(n, k) · p^k · (1 − p)^(n − k), where p is the probability of heads (0.5 for a fair coin). Choose exactly, at least, or at most k to sum the binomial probabilities, and expect n·p heads on average.
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, so the chance of exactly k heads is P(X = k) = C(n, k) · p^k · (1 − p)^(n − k), where C(n, k) is the binomial coefficient "n choose k". Enter the number of flips n, the target heads k, the per-flip head probability p, and a mode.
The "exactly" mode returns P(X = k). The "at least" mode sums the probability mass for k, k+1, …, n, and the "at most" mode sums from 0 up to k. A useful shortcut is P(at least 1 head) = 1 − (1 − p)^n. The expected (mean) number of heads is n·p and the variance is n·p·(1 − p). The bar chart shows the full probability mass function with the bars included by your chosen mode highlighted.
Inputs must satisfy n ≥ 1, 0 ≤ k ≤ n, and 0 ≤ p ≤ 1; otherwise no result is shown. Probabilities are computed with a numerically stable recurrence so large n stays accurate. With p = 0 the coin never shows heads (all mass at 0), and with p = 1 it always does (all mass at n).
Frequently asked questions
It is C(10, 5) · 0.5^10 = 252/1024 ≈ 24.6%, the most likely single outcome for a fair coin tossed 10 times.
Set P(head) to the coin's true head probability, for example 0.6 for a coin that lands heads 60% of the time. The binomial formula uses this p directly.
Use the "at least" mode with k = 1, or compute 1 − (1 − p)^n. For a fair coin and n = 3 flips that is 1 − 0.5^3 = 87.5%.
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-calculator
TG we-Calculate Editorial Team. "Coin Flip Probability Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/coin-flip-probability-calculator.
TG we-Calculate Editorial Team, "Coin Flip Probability Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/coin-flip-probability-calculator
@misc{wecalculate_coin_flip_probability_calculator, title = {Coin Flip Probability Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/coin-flip-probability-calculator}}, year = {2026}, note = {TG we-Calculate} }
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