Coin Flip Probability Calculator
The probability of getting exactly k heads in n fair coin flips is C(n,k) ÷ 2ⁿ; for example, exactly 5 heads in 10 flips is 24.6%, and 3 heads in a row is 12.5%.
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Calculator (n, k, "exactly / at least / at most")
The coin flip probability calculator answers questions like what the chance is of 7 heads in 10 flips. Enter the number of flips (n) and the number of heads (k), then choose exactly, at least or at most. Exactly 7 heads in 10 flips is 11.72%. At least one heads in 3 flips is 87.5%, and in 10 flips it is 99.90%. At most 2 heads in 3 flips is also 87.5%, since only the all heads outcome is left out. The calculator runs in your browser and shows the percentage, so you can drop it straight into a homework answer or settle a quick argument with a friend.
The formula (binomial)
The chance of exactly k heads in n fair coin flips is C(n,k) divided by 2 to the power n. The 2 to the power n part counts every possible sequence of heads and tails, and C(n,k), read as n choose k, counts the sequences that contain exactly k heads. For 3 flips there are 8 sequences, and 3 of them have exactly 2 heads, so the answer is 3/8, or 37.5%. For 10 flips there are 1,024 sequences, and 252 of them have exactly 5 heads, which gives 24.61%. This is the binomial distribution. It works because every flip is independent and each side always has a 1/2 chance.
Flipping a coin 3 times: all 8 outcomes
When you flip a coin 3 times there are 8 equally likely outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH and TTT. Each one has a 1 in 8 chance, or 12.5%. Count the heads and the pattern is easy to see. All 3 heads appears in one outcome (12.5%). Exactly 2 heads appears in three outcomes (37.5%), and exactly 1 heads is also 37.5%. No heads at all is 12.5%, so the chance of at least one heads in 3 flips is 87.5%. The same list answers the classic textbook question of what can happen when a coin is tossed three times.
10 flips chart
In 10 coin flips, 5 heads is the single most likely result at 24.61%, which still means you get something else about three times out of four. The chart shows every count from 0 to 10. 4 heads and 6 heads are 20.51% each, 3 and 7 are 11.72%, 2 and 8 are 4.39%, 1 and 9 are 0.98%, and 0 or 10 heads are 0.10% each. Add up the middle and 4 to 6 heads covers about 66% of all results. A split like 7 heads and 3 tails feels lopsided, yet it shows up more than one time in nine.
Streaks: k in a row
The chance of the same specific side k times in a row is 1 in 2 to the power k. Two heads in a row is 25% (1 in 4), three is 12.5% (1 in 8), five is 3.13% (1 in 32) and ten is 0.098% (1 in 1,024). If you do not care which side, only that every flip matches the first one, the chance doubles: two of a kind is 1 in 2, three is 1 in 4, five is 1 in 16 and ten is 1 in 512. Streaks somewhere inside a longer run of flips are far more common than these numbers suggest, as the 100 flip section shows.
100 flips: what's normal
In 100 flips you will usually land close to 50 heads, but rarely right on it. Exactly 50 heads happens 7.96% of the time. Anything from 45 to 55 heads covers 72.87% of results, and 40 to 60 covers 96.48%. Exactly 40 heads, or exactly 60, is 1.08%. Streaks are part of the picture too: a run of at least 5 of the same side somewhere in 100 flips shows up 97.2% of the time, at least 6 shows up 80.7% of the time, and at least 7 shows up 54.2% of the time. So if your 100 flips end 46 to 54 with a streak of seven, nothing is wrong with the coin.
The coin has no memory (each flip is independent)
A coin has no memory, so past flips never change the next one. After five tails in a row, the next flip is still 50% heads and 50% tails. The feeling that heads is due is a very common thinking trap, and it is wrong for a fair coin. Heads and tails even out over thousands of flips only because an early streak gets diluted by everything that follows, not because the coin corrects itself. That is also why the formulas on this page simply multiply: each flip is a fresh 1/2. The same holds for our digital coin, where every flip is computed on its own and tells you nothing about the next one.
Exactly k heads in n flips
| Heads | 3 flips | 5 flips | 10 flips |
|---|---|---|---|
| 0 | 12.5% | 3.13% | 0.10% |
| 1 | 37.5% | 15.63% | 0.98% |
| 2 | 37.5% | 31.25% | 4.39% |
| 3 | 12.5% | 31.25% | 11.72% |
| 4 | 15.63% | 20.51% | |
| 5 | 3.13% | 24.61% | |
| 6 | 20.51% | ||
| 7 | 11.72% | ||
| 8 | 4.39% | ||
| 9 | 0.98% | ||
| 10 | 0.10% |
Same side k times in a row
| k | Specific side (e.g. heads) | Any side, counted from the first flip |
|---|---|---|
| 2 | 1 in 4 (25%) | 1 in 2 |
| 3 | 1 in 8 (12.5%) | 1 in 4 |
| 5 | 1 in 32 (3.13%) | 1 in 16 |
| 10 | 1 in 1,024 (0.098%) | 1 in 512 |
| 15 | 1 in 32,768 | 1 in 16,384 |
| 20 | 1 in 1,048,576 | 1 in 524,288 |
100 flips: what's normal
| Result | Chance |
|---|---|
| Exactly 50 heads | 7.96% |
| 45 to 55 heads | 72.87% |
| 40 to 60 heads | 96.48% |
| Exactly 40 (or exactly 60) heads | 1.08% |
Longest run somewhere in 100 flips
| A run of at least… | Chance |
|---|---|
| 5 | 97.2% |
| 6 | 80.7% |
| 7 | 54.2% |
| 8 | 31.5% |
| 10 | 8.7% |
At least one heads in n flips
| 2 | 75% |
|---|---|
| 3 | 87.5% |
| 5 | 96.88% |
| 10 | 99.90% |
FAQ
What is the probability of getting heads 3 times in a row?
It is 1 in 8, or 12.5%. Each flip is independent with a 1/2 chance, so three heads in a row is 1/2 × 1/2 × 1/2. When you flip a coin 3 times, the chance of exactly 2 heads is 37.5% and of at least one heads is 87.5%. Five heads in a row is 1 in 32, and ten in a row is 1 in 1,024.
What happens if you flip a coin 100 times?
You will usually get close to 50 heads, but rarely exactly 50. The chance of exactly 50 heads in 100 flips is about 8%. The chance of 45 to 55 heads is about 73%, and of 40 to 60 heads about 96%. Streaks are normal too: there is a 54% chance of seeing seven or more of the same side in a row somewhere in 100 flips.