Binomial Probability: Coin Flips, Free Throws, and Defect Counts
October 21, 2026
Many real situations involve repeating the same yes/no trial several times: flipping a coin, a basketball player shooting free throws, a machine producing items that are either defective or fine, a customer clicking or not clicking an ad. When the trials are independent and each has the same probability of success, the number of successes follows a binomial distribution. Its formula gives the probability of exactly k successes in n trials.
The formula is P(k) = C(n, k) × p^k × (1 − p)^(n−k), where C(n, k) = n! ÷ (k!(n − k)!) counts the number of ways to arrange k successes among n trials.
Skip the math and get your answer instantly:
Open the Permutation & Combination Calculator →Worked Example: Coin Flips
What is the chance of exactly 3 heads in 10 flips of a fair coin? C(10, 3) = 120, p^k = 0.5³, and (1 − p)^(n−k) = 0.5⁷. P = 120 × 0.5¹⁰ = 120 ÷ 1,024 = 0.1172, or 11.7%. Exactly 5 heads, the most likely single outcome, is C(10, 5) × 0.5¹⁰ = 252 ÷ 1,024 = 24.6%.
The probability of at least 8 heads is the sum for 8, 9, and 10: (45 + 10 + 1) ÷ 1,024 = 56 ÷ 1,024 = 5.5%. That is the kind of result that gets labeled unusual in a test of fairness.
| Heads in 10 flips | Ways C(10,k) | Probability |
|---|---|---|
| 0 | 1 | 0.10% |
| 2 | 45 | 4.39% |
| 3 | 120 | 11.72% |
| 5 | 252 | 24.61% |
| 8 | 45 | 4.39% |
| 10 | 1 | 0.10% |
Worked Example: Free Throws and Defects
A basketball player makes 80% of free throws. The chance she makes exactly 4 of 5 is C(5, 4) × 0.8⁴ × 0.2 = 5 × 0.4096 × 0.2 = 0.4096, or 41%. The chance she makes all 5 is 0.8⁵ = 32.8%.
A process has a 2% defect rate. In a sample of 20, the chance of exactly 2 defectives is C(20, 2) × 0.02² × 0.98¹⁸ = 190 × 0.0004 × 0.6951 = 5.3%. The chance of none is 66.8% and of at least one is 33.2%.
Mean and Standard Deviation
The expected number of successes is n × p, and the standard deviation is √(n p (1 − p)). For 100 flips of a fair coin, the mean is 50 and the standard deviation is √(100 × 0.25) = 5. Therefore, results between 40 and 60 heads (within two standard deviations) happen about 95% of the time. Getting 70 or more heads would be extremely unlikely, a sign of a biased coin.
For large n, the binomial distribution looks like a normal bell curve, which allows quick approximations.
When the Binomial Applies
There must be a fixed number of trials; each trial has two outcomes; the probability of success is the same each time; and trials are independent. Drawing cards without replacement violates independence (use the hypergeometric distribution instead). Shooting percentages may change with fatigue or pressure, so the model is only an approximation. Check the assumptions before relying on the result.
Using a Calculator or Spreadsheet
Spreadsheets provide BINOM.DIST(k, n, p, FALSE) for the exact probability and TRUE for the cumulative probability of at most k successes. For 'at least k', use 1 minus the cumulative probability of k − 1.
Frequently Asked Questions
What is the binomial probability formula?
P(k) = C(n, k) × p^k × (1 − p)^(n − k).
What is the chance of exactly 5 heads in 10 flips?
About 24.6%.
What are the assumptions of a binomial model?
Fixed number of trials, two outcomes, constant success probability, and independent trials.
How do I find 'at least' probabilities?
Add the probabilities from k up to n, or subtract the cumulative probability below k from 1.
What is the mean of a binomial distribution?
n × p.