Binomial Distribution Calculator
Calculate binomial probabilities for exactly, at most, at least, less than, greater than, between, and outside a range. Get the mean, variance, standard deviation, probability table, inverse cutoff, formula, and steps.
Binomial probability calculator
Find exact, cumulative, tail, range, and outside-range probabilities.
Inverse binomial calculator
Find the smallest number of successes x for which P(X ≤ x) reaches your target probability.
This discrete inverse uses the first integer cutoff whose cumulative probability meets or exceeds the target.
Binomial probability distribution table
P(X = k) for each possible number of successes.
| k | P(X = k) | Distribution |
|---|---|---|
| 0 | 0.0977% | |
| 1 | 0.9766% | |
| 2 | 4.3945% | |
| 3 | 11.7188% | |
| 4 | 20.5078% | |
| 5 | 24.6094% | |
| 6 | 20.5078% | |
| 7 | 11.7188% | |
| 8 | 4.3945% | |
| 9 | 0.9766% | |
| 10 | 0.0977% |
Step-by-step calculation
- Identify the inputs: n = 10 trials, p = 0.5, and x = 5 successes.
- Use the binomial PMF: P(X = x) = C(n, x)pˣ(1 − p)ⁿ⁻ˣ.
- Calculate the selected probability: P(X = 5) = 24.6094%.
- For cumulative questions: add the relevant exact probabilities, such as P(X ≤ 5) = 62.3047%.
- Interpret the result: the selected probability is 24.6094% under these assumptions.
How to use the binomial distribution calculator
- Enter the number of trials n.
- Enter the probability of success p as a decimal from 0 to 1.
- Enter the success count x. For a range, enter the upper value b too.
- Read the exact, cumulative, tail, range, and outside-range probabilities.
- Use the distribution table and step-by-step calculation to check your work.
Binomial distribution conditions
A problem follows a binomial distribution when it has a fixed number of trials, exactly two modeled outcomes per trial, independent trials, and the same probability of success for every trial.
How to use the binomial formula
Here, C(n, x) is the number of ways to choose x successes from n trials. Enter n, p and x in the calculator above to evaluate the formula.
Expected value and mean of a binomial distribution
The expected value, or mean, is μX = np. For n = 20 and p = 0.4, μX = 20 × 0.4 = 8.
How to find the standard deviation of a binomial distribution
First calculate the variance with σ² = np(1 − p), then take its square root: σ = √[np(1 − p)]. For n = 20 and p = 0.4, the standard deviation is √(20 × 0.4 × 0.6) ≈ 2.191.
What is BinomCDF?
BinomCDF is a calculator function for the cumulative binomial probability. In common calculator notation, binomcdf(n, p, x) represents P(X ≤ x). For example, binomcdf(10, 0.5, 3) gives the probability of 3 or fewer successes.
For an at least probability, use the complement: P(X ≥ 4) = 1 − P(X ≤ 3). For a range, P(3 ≤ X ≤ 7) = P(X ≤ 7) − P(X ≤ 2).
Read the full guide to using BinomCDF →
Binomial probability experiment examples
- Exactly: n = 20, p = 0.3, find P(X = 5).
- At most: n = 11, p = 0.5, find P(X ≤ 4).
- At least: n = 10, p = 0.8, find P(X ≥ 6).
- Between: n = 20, p = 0.4, find P(5 ≤ X ≤ 10).
Worked example
Suppose a student guesses on 10 true/false questions, so n = 10 and p = 0.5. The probability of exactly 7 correct answers is calculated with P(X = 7) = C(10,7)(0.5)⁷(0.5)³, which is approximately 11.7188%.
Binomial probability examples
- Exactly x: P(X = x)
- At most x: P(X ≤ x)
- At least x: P(X ≥ x)
- Less than x: P(X < x)
- More than x: P(X > x)
- Between a and b: P(a ≤ X ≤ b)
Frequently asked questions
What is a binomial distribution?
A binomial distribution models the number of successes in a fixed number of independent trials when each trial has the same probability of success.
What does n mean in a binomial distribution?
n is the fixed number of trials, such as 10 coin flips, 20 free throws, or 50 quality-control checks.
What does p mean in a binomial distribution?
p is the probability of success on each trial. Enter it as a decimal between 0 and 1, so 25% is 0.25.
What is x in a binomial distribution?
x is the number of successes being evaluated. For a binomial model, x must be an integer from 0 through n.
What is the binomial probability formula?
For exactly x successes, P(X = x) = C(n,x)p^x(1-p)^(n-x).
What is the difference between P(X ≤ x) and P(X < x)?
P(X ≤ x) includes x, while P(X < x) excludes x. Likewise, P(X ≥ x) includes x and P(X > x) excludes x.
When should I use a binomial distribution?
Use it when there is a fixed number of trials, two outcomes per trial, independent trials, and the same probability of success on every trial.