Coin flip guide
Coin Flip Probability Explained
Start by naming the event: exactly two heads, at least one head, or a particular sequence. These similar-sounding questions have different answers.
List the outcomes first
Two independent fair flips have four equally likely outcomes: HH, HT, TH, and TT. Exactly one head occurs in HT and TH, so its probability is 2/4 = 50%. A specific sequence such as HT has probability 1/4 = 25%.
Count heads without fixing their positions
For n fair flips and k heads, P(X = k) = C(n,k)/2ⁿ. Exactly two heads in four flips can appear as HHTT, HTHT, HTTH, THHT, THTH, or TTHH: six of sixteen outcomes, giving 37.5%.
Use a complement
At least one head in four flips means every sequence except TTTT. Its probability is 1 − 1/16 = 93.75%. Counting the excluded event is easier than listing all fifteen successful sequences.
Check the assumptions
These examples assume independent trials with equal side probabilities. For a weighted coin the exact-count formula is C(n,k)pᵏ(1−p)ⁿ⁻ᵏ. Our calculator fixes p at 0.5; it does not estimate a physical coin’s bias.
| Heads | Sequences | Probability |
|---|---|---|
| 0 | 1 | 6.25% |
| 1 | 4 | 25% |
| 2 | 6 | 37.5% |
| 3 | 4 | 25% |
| 4 | 1 | 6.25% |
Sources and further reading
Worked examples use the independent fair-coin model unless a section explicitly discusses physical tossing.