Coin flip probability formula
For n independent flips, k heads, and a heads probability p, the binomial formula is:
P(X = k) = C(n, k) × pk × (1 - p)n-k
For a fair coin, p is 0.5. C(n, k) counts how many ordered sequences contain exactly k heads.
Exact, at least, and at most
- Exactly k heads
- Only outcomes containing the requested number of heads.
- At least k heads
- The requested number or any larger total.
- At most k heads
- The requested number or any smaller total.
Common fair-coin examples
- Heads on one flip: 50%
- Two heads in two flips: 25%
- Exactly one head in two flips: 50%
- At least one head in three flips: 87.5%
- Exactly five heads in ten flips: about 24.61%
Compare the calculation with observed outcomes using the multiple coin flipper.
Probability questions
Do previous coin flips affect the next one?
No. Independent flips have no memory. The next fair flip remains 50% heads and 50% tails.
Can this calculate a biased coin?
Yes. Change the heads chance from 50% to any value from 0% through 100%.
Why is half heads not guaranteed?
Expected value describes an average across many repeated experiments. It does not require a particular small batch to be balanced.