Coin Flip Probability Calculator
The coin flip probability calculator computes the probability of getting exactly k heads in n flips of a coin. For a fair coin, the formula is P = C(n,k) * (0.5)^n. For a biased coin, use P = C(n,k) * p^k * (1-p)^(n-k). The calculator also shows cumulative probability: P(X <= k), the probability of getting k or fewer heads. Enter the number of flips, number of heads, and probability of heads (default 0.5 for a fair coin).
Formula
P(X = k) = C(n,k) * p^k * (1-p)^(n-k)
where C(n,k) = n! / (k! * (n-k)!)
Coin flip probability calculator: frequently asked questions
What is the probability of getting exactly k heads?
For a fair coin, the probability is C(n,k) * (0.5)^n, where n is the number of flips and k is the number of heads. For example, the probability of exactly 5 heads in 10 flips is C(10,5) * (0.5)^10 ≈ 24.61%.
What if the coin is biased?
If the coin is biased (probability of heads not 0.5), use P = C(n,k) * p^k * (1-p)^(n-k), where p is the probability of heads. This is the binomial probability formula.
What is cumulative probability?
Cumulative probability P(X <= k) is the probability of getting k or fewer heads. For example, P(X <= 3) includes outcomes with 0, 1, 2, or 3 heads.
Why is 50/50 the most likely outcome?
For a fair coin flipped an even number of times, the number of ways to get exactly half heads equals the number of combinations C(n, n/2). This is the largest binomial coefficient for that n.
Can I model a biased coin?
Yes. Enter the probability of heads. For example, a coin that shows heads 60% of the time has p = 0.6. The calculator will use the binomial formula with your custom probability.
Official sources
- Wikipedia: Binomial Distribution.
- Wolfram MathWorld: Binomial Distribution.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.