Coin Flip Probability Calculator
Getting exactly 5 heads in 10 fair flips has only a 24.61% chance, even though 5 is the most likely single count. This coin flip probability calculator computes the exact binomial probability of any number of heads in any number of flips, plus the at-least and at-most cumulative chances. Ten flips rarely land the tidy 5-5 split intuition promises.
Quick answer
Each fair flip is 50/50, and n flips produce 2^n equally likely sequences.
Exactly this many heads
24.6094%
At least
62.3047%
At most
62.3047%
About 1 in
4.1
What this tells you
- •Each fair flip is 50/50, and n flips produce 2^n equally likely sequences.
- •The chance of exactly k heads is the number of orderings, C(n, k), divided by 2^n.
- •At-least and at-most probabilities add up the exact chances across a range of counts.
- •More flips concentrate results near half heads, but the chance of exactly half actually shrinks.
How to Use
- 1Enter the total number of flips.
- 2Enter the number of heads you care about.
- 3Read the exact probability, plus the at-least and at-most cumulative chances.
- 4The 1-in-N figure translates small probabilities into plainer language.
How It Works
Formula
P(exactly k) = C(n, k) / 2^nC(n, k) counts how many orderings place k heads among n flips. For 5 heads in 10 flips, C(10, 5) = 252 orderings out of 2^10 = 1024 total, giving 24.61%. The calculator works in log space, so it stays exact for thousands of flips where direct factorials overflow.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Exactly 5 heads in 10 flips
252 of the 1024 possible sequences have exactly 5 heads.
10 heads in a row
Only one sequence of 1024 is all heads.
At least 60 heads in 100 flips
A 60/40 split in 100 flips is already rare enough to raise an eyebrow about the coin.
Exactly k Heads in 10 Flips
The full probability distribution for 10 fair flips.
| Heads | Probability |
|---|---|
| 0 or 10 | 0.10% each |
| 1 or 9 | 0.98% each |
| 2 or 8 | 4.39% each |
| 3 or 7 | 11.72% each |
| 4 or 6 | 20.51% each |
| 5 | 24.61% |
Common mistakes
- Expecting exactly half heads to be likely. It is only the single most likely count, 24.61% for 10 flips and shrinking as flips grow.
- Believing a streak changes the next flip. After 9 heads, the 10th is still 50/50. The 1-in-1024 rarity belonged to the whole sequence in advance.
- Confusing exactly with at least. Exactly 5 in 10 is 24.61%, at least 5 is 62.3%, very different questions.
- Adding 50% per flip. The chance of at least one head in two flips is 75%, not 100%.