How this calculator works
The number of heads follows a binomial distribution when flips are independent and share a constant chance of heads. The combination factor counts the different orders producing a particular head count. Range endpoints are inclusive. A prescribed sequence has a different probability because its order is fixed. Zero flips always produces zero heads.
Formula
P(H = k) = C(n,k)p^k(1 - p)^(n - k). Add these probabilities across the requested counts. E(H) = np; Var(H) = np(1 - p).
Worked example
Exactly two heads in three fair flips can occur as HHT, HTH or THH. Each has probability 1/8, so the total is 3/8 = 37.5%. At least three heads in four fair flips has probability 5/16.
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Frequently asked questions
Does this calculate consecutive heads?
No. Heads can occur in any order. Use the coin toss streak calculator for a consecutive run.
Are at least and at most inclusive?
Yes. At least three includes three, and at most three includes three.
Can the coin be biased?
Yes. Enter any fixed chance from 0% through 100%. The flips must still be independent for the binomial model.