Binomial Probability Calculator
The binomial distribution models the number of successes in n independent trials that each succeed with the same probability p.
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How to use this calculator
- Enter the number of trials n (e.g. 10 coin flips), the number of successes k you are interested in, and the probability p of success on a single trial (as a decimal from 0 to 1, so 50% = 0.5).
- Read P(X = k) for the exact probability, or use the cumulative rows — P(X ≤ k), P(X ≥ k), P(X < k), P(X > k) — for 'at most', 'at least', 'fewer than' and 'more than' style questions.
- Check the mean (n × p) and standard deviation to see where k sits relative to the expected number of successes.
How it works
The binomial distribution models the number of successes in n independent trials that each succeed with the same probability p. The chance of exactly k successes is P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) is the number of ways to choose which k of the n trials are the successes.
Cumulative results add up the individual probabilities: P(X ≤ k) sums P(X = 0) through P(X = k), and 'at least' and 'more than' are found by subtracting the appropriate cumulative sum from 1. The mean of a binomial is n × p and the standard deviation is √(n × p × (1 − p)). The calculator uses a stable multiplicative formula for C(n, k) so large trial counts do not overflow.
Worked example
Exactly 3 heads in 10 fair coin flips. Flip a fair coin 10 times, so n = 10, p = 0.5, and you want exactly k = 3 heads. The number of ways to place 3 heads is C(10, 3) = 120, and each specific pattern of 3 heads and 7 tails has probability 0.5^3 × 0.5^7 = 0.5^10 ≈ 0.000977. Multiplying gives P(X = 3) = 120 × 0.000977 = 0.117188, so there is about an 11.7% chance of seeing exactly 3 heads. The calculator also reports P(X ≤ 3) = 0.171875, P(X ≥ 3) = 0.945313, a mean of 5, and a standard deviation of about 1.581139.
Common mistakes
- Entering p as a percentage instead of a decimal — a 25% chance is 0.25, not 25. Values above 1 are rejected.
- Confusing 'exactly k' with 'at least k'. P(X = k) is a single bar of the distribution; use the P(X ≥ k) row for 'k or more'.
- Setting k greater than n. You cannot have more successes than trials, so k must be a whole number between 0 and n.
Frequently asked questions
What is the probability of getting exactly k successes in n trials?
Use P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). Enter n, k and p above and read the P(exactly k) row. For example, exactly 3 heads in 10 fair coin flips is about 0.1172, or 11.7%.
How do I work out 'at least' or 'at most' probabilities?
For 'at most k' successes read P(X ≤ k). For 'at least k' read P(X ≥ k), which equals 1 minus the probability of fewer than k successes. The calculator shows all four cumulative cases so you don't have to sum the terms by hand.
When can I use the binomial distribution?
When you have a fixed number of independent trials, each with only two outcomes (success or failure) and the same success probability p on every trial. Coin flips, pass/fail tests and defect counts in a fixed sample all fit this model.
Related tools
- Combinations Calculator
- Normal Distribution Calculator
- Permutations Calculator
- Standard Deviation Calculator
- Z-Score Calculator
- Confidence Interval Calculator
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