Coin flip probability calculator

The formula

The number of heads X in n independent flips, each landing heads with probability p, is binomial:

P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, C(n, k) = n! ÷ (k! (n − k)!)

C(n, k) counts the sequences with exactly k heads, and pᵏ(1 − p)ⁿ⁻ᵏ is the probability of any one of them. “At least”, “at most” and “between” add these terms over the counts in the range. For a fair coin every sequence has probability (½)ⁿ, so the answer is simply the number of favourable sequences divided by 2ⁿ.

A worked example

The default asks for the chance of at least 7 heads in 10 flips of a fair coin. Add the counts of sequences with 7, 8, 9 and 10 heads:

C(10, 7) + C(10, 8) + C(10, 9) + C(10, 10) = 120 + 45 + 10 + 1 = 176

P(X ≥ 7) = 176 ÷ 1024 = 0.1719

So about 17% of the time, 10 fair flips give 7 or more heads. On its own, exactly 7 heads has probability 120 ÷ 1024 = 0.1172. By symmetry, at most 3 heads is also 176 ÷ 1024.

Now make the coin biased with P(heads) = 0.6. The same question gives 0.3823, more than double, because each extra head is now more likely than a tail. The expected number of heads moves from 5 to 6.

Interpreting the result

The chart shows every possible count of heads, with the bars in your event drawn solid. Its peak sits at n × p and its width is set by the standard deviation √(n p (1 − p)). A probability below about 0.05 means the outcome would be surprising for a coin with the p you entered; that is the logic of a one-sided binomial test. If you are checking whether a real coin is fair, 7 heads in 10 is nowhere near enough evidence: it happens one time in six.

For large n the binomial is close to a normal curve with the same mean and SD, but the calculator sums the exact binomial terms, so there is no continuity correction to worry about.

Mean and spread of the number of heads

The cells under the result give the expected number of heads, n × p, and its standard deviation, √(n p (1 − p)). For 10 fair flips that is 5 ± 1.58. For 100 fair flips it is 50 ± 5, so 60 or more heads is two standard deviations high and has probability 0.028. Scaling up makes the proportion of heads steadier while the count itself spreads out further, which is the law of large numbers at work.

Common mistakes

• Forgetting the C(n, k) term. The chance of 2 heads in 3 flips is 3 × ⅛ = ⅜, not ⅛, because HHT, HTH and THH all count.
• Mixing up “at least 7” and “more than 7”. At least 7 includes 7; more than 7 starts at 8.
• Adding probabilities across separate sets of flips as if they were one set. The chance of at least one head in two separate 3-flip rounds is not ⅞ + ⅞.