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- Finds the probability of exactly x successes in n independent trials with success chance p.
- Also gives cumulative probabilities: at most, fewer than, at least and more than x.
- Shows the mean, variance and standard deviation of the distribution.
- Lists the full probability table for up to 30 trials.
How to use the binomial distribution calculator
Enter the number of trials (n)
For example 10 coin flips or 20 free throws.
Enter the probability of success (p)
A value from 0 to 1. A fair coin is 0.5, a 1-in-4 chance is 0.25.
Enter the number of successes (x)
The count you want the probability for.
Read all the probabilities
Exact, at most, at least, fewer than and more than are all shown together.
How binomial probabilities are calculated
Binomial probability formula
P(X = x) = C(n, x) × pˣ × (1 − p)ⁿ⁻ˣ
C(n, x) = n! ÷ (x! × (n − x)!)
C(n, x) counts the number of ways to arrange x successes among n trials.
Cumulative probabilities and summary
P(X ≤ x) = P(X = 0) + P(X = 1) + … + P(X = x)
P(X ≥ x) = 1 − P(X < x)
Mean = n × p, Variance = n × p × (1 − p)
Example 1: 6 heads in 10 coin flips
n = 10, p = 0.5, x = 6.
- P(X = 6) 0.205078
- P(X ≤ 6) 0.828125
- P(X ≥ 6) 0.376953
- Mean 5
- Standard deviation 1.581139
About a 20.5% chance of exactly 6 heads.
Example 2: a 30% chance, 5 tries
n = 5, p = 0.3, x = 2.
- C(5, 2) 10
- P(X = 2) 10 × 0.3² × 0.7³ = 0.3087
- P(X ≥ 2) 0.47178
Even with only a 30% chance each time, getting at least 2 successes in 5 tries happens almost half the time.
Example 3: zero successes
n = 20, p = 0.25, x = 0.
- P(X = 0) 0.75²⁰ ≈ 0.003171
The chance of missing every time is small, which is why repeated attempts usually succeed at least once.
Where a binomial calculator helps
Statistics classes
Check binompdf and binomcdf answers without a graphing calculator.
Quality control
Find the chance of a set number of faulty items in a batch.
Sports and games
Work out the odds of making a number of shots, wins or rolls.
Marketing and testing
Estimate how many people out of a sample will respond or convert.
Use it when there is a fixed number of trials, each trial has only two outcomes, the trials are independent, and the chance of success stays the same. For large n, the distribution looks like a normal curve, so you can also use the z-score calculator. Turn a probability into a percent with the percentage calculator, or build an interval for a proportion with the confidence interval calculator.
Binomial distribution calculator FAQ
What is a binomial distribution?
It describes the number of successes in a fixed number of independent trials, where each trial has the same chance of success.
What is the difference between binompdf and binomcdf?
binompdf gives the probability of exactly x successes. binomcdf gives the probability of x or fewer. This calculator shows both, plus the other tail.
How do I find the probability of at least x successes?
Use P(X ≥ x). It equals 1 minus the probability of fewer than x successes, and the calculator shows it directly.
What are the mean and variance of a binomial distribution?
The mean is n × p and the variance is n × p × (1 − p). The standard deviation is the square root of the variance.
Can p be a percentage?
Enter it as a decimal. For 30%, type 0.3.
How large can n be?
Up to 100,000 trials. The full table is shown for 30 trials or fewer.