HesapHanem

Coin Flip Probability Calculator

Enter the number of flips and the number of heads.

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%.

Worked example

Flip a fair coin 10 times and ask for 7 heads. Ways to pick which 7 of the 10 flips are heads: C(10,7) = 120. All outcomes: 2¹⁰ = 1,024. Exactly 7 heads: 120 ÷ 1,024 = 11.7188%. For at least 7 heads add the cases of 7, 8, 9 and 10 heads: 120 + 45 + 10 + 1 = 176, and 176 ÷ 1,024 = 17.1875%. The expected number of heads is 10 ÷ 2 = 5.00.

Try this example →

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.

About this calculator

Flipping a fair coin n times is the textbook example of a binomial distribution: each flip is an independent trial with a 50% chance of heads, and the probability of getting exactly k heads is given by P(k) = C(n,k) × 0.5ⁿ, where C(n,k) counts the number of ways to choose which k flips land heads.

This calculator computes that exact probability along with the cumulative probability of getting at least k heads, and the expected number of heads over n flips (always n/2 for a fair coin). It’s a go-to example in introductory probability and statistics courses, and the same math underlies fairness checks in game design and simple demonstrations of random processes in decision theory.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    Landing exactly on the average is less likely than you might expect, and it gets less likely as flips increase. Exactly 5 heads in 10 flips is 24.6094%, while exactly 500 in 1,000 flips is only 2.5225%, although both are the single most likely count. Post the flips and heads count you were wondering about.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators