Coin Flip Probability Calculator

How it is calculated

The coin is assumed fair, with a 50% chance of heads on every independent flip. For n flips and k heads the number of ways to choose which flips are heads is C(n,k) = n! ÷ (k! × (n−k)!), and the number of all outcomes is 2ⁿ. So P(exactly k) = C(n,k) ÷ 2ⁿ and P(at least k) is the sum of C(n,i) ÷ 2ⁿ for i from k to n. The expected number of heads is n ÷ 2.

The terms are computed in logarithms, so large n works: 2,000 flips are fine, and the limit is 1,000,000 flips. Both n and k must be whole numbers, n at least 0 and k between 0 and n; anything else (k above n, a decimal such as 10.5, n over 1,000,000) shows no result. Percentages are shown with four decimals, so tiny probabilities can display as 0.0000%.

Common mistakes

• Reading "exactly k" as "at least k". For 7 heads in 10 flips they are 11.7188% and 17.1875%, so check which line answers your question.
• Expecting the single most likely outcome to be likely. Even the most common result, 5 heads in 10 flips, happens only about a quarter of the time.
• Believing that after a run of tails heads is "due". Each flip is independent, so past results never change the probabilities.
• Using the tool for a biased or weighted coin. It always assumes 50% heads, so it does not apply to an event with other odds.
• Entering a proportion such as 0.7 or a percentage as k. The field is a count of heads, a whole number between 0 and n.