Coin Flip Probability Calculator
Find the exact probability of getting a chosen number of heads in a set of coin flips. Useful for statistics homework, game design, or verifying fair-coin assumptions.
About this calculator
This calculator uses the binomial probability formula: P(k heads in n flips) = C(n,k) × (0.5)^n, where C(n,k) = n! / (k! × (n−k)!) is the number of combinations. Because a fair coin has equal probability of heads or tails (p = 0.5), each specific sequence of k heads and (n−k) tails has probability (0.5)^n. Multiplying by the number of ways to arrange those outcomes gives the total probability. The result tells you how likely it is to observe exactly k heads out of n flips — not at least k, but precisely k. This is the foundation of binomial distributions used throughout statistics and probability theory.
How to use
Suppose you flip a coin 5 times and want to know the probability of getting exactly 3 heads. Enter Flips = 5 and Heads = 3. The calculator computes C(5,3) = 5!/(3!×2!) = 10, then multiplies by (0.5)^5 = 0.03125. Result: 10 × 0.03125 = 0.3125, or 31.25%. That means roughly 1 in 3 five-flip sessions will yield exactly 3 heads.
Frequently asked questions
What is the probability of getting heads every time in multiple coin flips?
Getting heads on every flip means k = n, so C(n,n) = 1 and the probability is simply (0.5)^n. For 5 flips that is (0.5)^5 = 3.125%. Each additional flip halves the probability, so streaks become exponentially rarer the longer they go.
How does the binomial formula work for coin flip probability?
The binomial formula counts all distinct sequences containing exactly k heads using the combination C(n,k), then multiplies by the probability of any single such sequence. For a fair coin every sequence of n flips has probability (0.5)^n, so the formula is C(n,k) × (0.5)^n. This cleanly separates 'how many ways' from 'how likely each way is'.
Why is the most likely number of heads equal to half the number of flips?
By symmetry, a fair coin has no bias toward heads or tails, so the distribution of outcomes is centred at n/2. The binomial distribution peaks at k = n/2 (or the nearest integer when n is odd) because there are more ways to arrange an equal split than any skewed split. This peak is called the mode of the distribution and it broadens as n grows.