What Is Binomial Probability?
The binomial distribution counts the successes in n independent trials that each have the same success probability p. The probability of exactly k successes is C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) is the number of ways to choose which trials succeed.
Enter n, p and k to get the exact probability and the cumulative probabilities, together with the mean np, the variance np(1 − p), the standard deviation and the most likely value. The worked steps show the binomial coefficient and each power, so you can follow or check a hand calculation.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as when a situation is binomial and how to find "at least" probabilities.
How to Use the Binomial Probability Calculator
- Enter the number of trials n, the target number of successes k and the success probability p.
- Choose which probability to feature: exactly, at most, fewer than, at least or more than k.
- Read the steps to see how the answer is built, and check the shaded bars in the chart.
- Use the table for neighbouring values of k.
Binomial Probability Formula
Binomial probability mass function
Binomial coefficient
Cumulative probability
Mean and variance
Binomial Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Number of trials | Fixed in advance |
| p | Success probability per trial | Same for every trial |
| k | Number of successes | 0 to n |
| C(n, k) | Binomial coefficient | n! ÷ (k!(n − k)!) |
| μ, σ² | Mean and variance | np and np(1 − p) |
Binomial Probability Worked Example
Using the example values (press “Load example” above to try them):
- Binomial coefficient: C(10, 3) = 10! ÷ (3! × 7!) = 120.
- p^k = 0.4^3 = 0.064; (1 − p)^(n − k) = 0.6^7 = 0.0279936.
- P(X = 3) = 120 × 0.064 × 0.0279936 = 0.21499085.
- Cumulative P(X ≤ 3) = Σ P(X = i) for i = 0 to 3 = 0.3822806; P(X ≥ 3) = 1 − P(X ≤ 2) = 0.83271025.
- Mean = np = 4, standard deviation = √(np(1 − p)) = 1.54919.
Result: P(X = 3) = 0.21499085; Mean μ = np = 4.
When is a situation binomial?
Drawing cards without replacement fails the last two conditions, so the exact model is hypergeometric. When the sample is under about 10% of the population, the binomial is a good approximation.
- A fixed number of trials n is decided in advance.
- Each trial has just two outcomes, success or failure.
- The probability of success p is the same on every trial.
- Trials are independent.
Translating words into probabilities
| Wording | Notation | Result to read |
|---|---|---|
| exactly 3 | P(X = 3) | Exact |
| at most 3, no more than 3, 3 or fewer | P(X ≤ 3) | At most |
| fewer than 3, less than 3 | P(X < 3) = P(X ≤ 2) | Fewer than |
| at least 3, 3 or more, no fewer than 3 | P(X ≥ 3) = 1 − P(X ≤ 2) | At least |
| more than 3, greater than 3 | P(X > 3) = 1 − P(X ≤ 3) | More than |
Normal and Poisson approximations
For large n the binomial is well approximated by a normal distribution with mean np and variance np(1 − p), provided np and n(1 − p) are both at least about 10. Apply a continuity correction of 0.5. When n is large and p is small, the Poisson distribution with mean np is a good approximation. This calculator computes the exact binomial values, so no approximation is needed.
Where Binomial Probability Is Used
Coin, dice and card problems
Probability of a given number of heads, sixes or hits.
Quality control
Probability of finding a given number of defects in a sample.
Surveys and trials
Probability of a given number of "yes" answers or recoveries.
Homework with working
Checking each step of a binomial probability.
Common Binomial Probability Mistakes to Avoid
- Convert percentages to decimals before entering p.
- For “at least one”, use 1 − P(X = 0).
- Check that the trials really are independent.
- Remember P(X ≥ k) is not the same as 1 − P(X ≤ k): subtract P(X ≤ k − 1).
Binomial Probability Terms and Definitions
- Success
- The outcome you are counting, not necessarily a good one.
- PMF
- Probability mass function: probability of an exact value.
- CDF
- Cumulative distribution function: probability of a value or less.
Binomial Probability FAQs
How do you calculate a binomial probability?
Use P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ: multiply the number of ways to choose k successes by the probability of one such arrangement.
What is the difference between binomial pdf and cdf?
The pdf (binompdf) gives the probability of exactly k successes. The cdf (binomcdf) gives the probability of k or fewer.
How do I find the probability of at least k successes?
Compute 1 − P(X ≤ k − 1). The result P(X ≥ k) is shown directly above.
What is the mean of a binomial distribution?
np, the number of trials multiplied by the success probability.
When can I use the normal approximation?
When np and n(1 − p) are both at least 10. Use a continuity correction.
Can the number of trials be large?
Yes, up to 100,000 here. The calculator uses stable log-based formulas for large n.
What if trials are not independent?
Then the binomial does not apply. For sampling without replacement from a small population use the hypergeometric distribution.