Coin Flip Probability Calculator

A step-by-step worked example

Take 10 flips and ask for exactly 7 heads. First count the outcomes. Ten flips give 2 to the power of 10, or 1,024, equally likely sequences. Next count the sequences with exactly 7 heads. That is C(10,7), the number of ways to choose which 7 of the 10 flips are heads, which works out to 10 x 9 x 8 divided by 3 x 2 x 1, or 120. Divide 120 by 1,024 and you get about 11.7%.

For "at least 7", add the counts for 7, 8, 9 and 10 heads: 120 + 45 + 10 + 1 = 176. Divide by 1,024 for about 17.2%. "At most 7" is everything except 8, 9 and 10 heads, so it is 1 minus 56 / 1,024, or about 94.5%. The calculator does this addition for you, for up to 1,000 flips.

What the inputs mean

In the first panel, n is the total number of flips and k is the number of heads you are asking about. Exactly means k heads and no other count. At least means k or more. At most means k or fewer. Because heads and tails are symmetric, the chance of k heads always equals the chance of k tails, so you can use the same numbers for tails.

In the streak panel, the number of flips is the length of the whole sequence and the streak length is the run of heads in a row you are asking about. The answer is the chance that at least one such run appears anywhere in the sequence. A run longer than your target counts too, so a run of eight heads also satisfies a target of seven.

How the streak odds are calculated

Streaks cannot be worked out with a single formula like C(n,k), because runs overlap and can occur anywhere. The calculator instead steps through the sequence one flip at a time, tracking the chance of being in a current run of length 0, 1, 2 and so on, up to the target. Each flip either extends the run with heads or resets it to zero with tails. Once a run reaches the target, that probability is moved to a "streak happened" total. The result is exact, not a simulation.

Some results: in 10 flips the chance of at least three heads in a row is about 50.8%. In 20 flips, four in a row is about 47.8%. In 100 flips, seven heads in a row is about 31.8%, but a run of seven of the same side, heads or tails, rises to about 54.2%. Keep that difference in mind, because the calculator counts heads only.

Waiting times and Penney's game

A related question is how many flips you need, on average, to see a given streak. For a run of r heads the answer is 2 to the power of r+1, minus 2. Three heads in a row takes 14 flips on average and five in a row takes 62. Patterns that look equally likely can have different waiting times. You wait on average 6 flips for heads-heads but only 4 for heads-tails, because a failed attempt at heads-tails still leaves you holding the first head. This idea is the basis of Penney's game, where the second player can always choose a three-flip pattern that is more likely to appear first.

Common mistakes

Confusing "exactly" with "at least" is the classic slip. The chance of exactly 60 heads in 100 flips is about 1.1%, but the chance of at least 60 is about 2.8%. Another is entering a value of k larger than n, which has no meaning. A third is reading a tiny percentage as zero. For long sequences the calculator can show values well under 0.1%, which is rare but still possible.

Frequently asked questions