Coin Flip Probability Calculator

Use the binomial probability formula: P(X = k) = C(n, k) × p^k × (1−p)^(n−k), where n is the total number of flips, k is the desired number of heads, and p is the probability of heads on a single flip. C(n, k) is the binomial coefficient, also written as n! / (k! × (n−k)!).

For a fair coin, the probability of heads on any single toss is 0.5 (50%). For multiple tosses, the binomial formula P(X = k) = C(n, k) × (0.5)^n gives the probability of getting exactly k heads in n flips.

Set n = 10, k = 8, and p = 0.5, then apply the binomial formula: C(10, 8) × (0.5)^8 × (0.5)^2 = 45 × 0.003906 ≈ 0.0439, or about 4.39%. You can verify this directly with the calculator above.

The easiest approach uses the complement rule: P(at least 1 head) = 1 − P(0 heads) = 1 − (0.5)^4 = 1 − 0.0625 = 0.9375, or 93.75%. Select 'At least X heads' with n = 4 and X = 1 in this calculator to confirm.

Using the binomial formula with n = 3, k = 2, p = 0.5: C(3, 2) × (0.5)^2 × (0.5)^1 = 3 × 0.25 × 0.5 = 0.375, or 37.5%.

'Exactly X heads' calculates the probability of getting precisely that number — no more, no less. 'At least X heads' sums the probabilities of getting X, X+1, X+2, … up to n heads, representing the chance of meeting or exceeding your target count.

Yes. Simply change the 'Probability of Heads (p)' field to any value between 0 and 1. For example, enter 0.6 if the coin lands heads 60% of the time. The calculator uses the full binomial formula and works for any valid probability.

Even though each flip has a 50% chance of heads, getting exactly 8 out of 16 is just one specific outcome among many possible distributions. The binomial formula accounts for all the ways 8 heads can appear across 16 flips, which gives a probability of about 19.64% — much less than 50%.