Binomial Probability Calculator With Table
This binomial probability calculator finds the chance of getting exactly k successes in n independent trials when each trial succeeds with probability p, like heads in 10 coin flips or defective parts in a batch of 50. Enter n, p and k to get P(X = k) plus every cumulative probability: at most k, fewer than k, at least k and more than k. You also get the mean np, the variance and standard deviation, the formula with your numbers plugged in, and a full probability table you can download as a CSV file.
Binomial Probability Calculator
Find P(X = k), P(X ≤ k), P(X ≥ k) and more for n trials with success probability p.
How to use the Binomial Probability Calculator
- Enter the number of trials n.
- Enter the probability of success p (decimal or %).
- Enter the number of successes k.
- Read the exact and cumulative probabilities.
Binomial formula
P(X = k) = C(n, k) · p^k · (1 − p)^(n − k) · mean = np · SD = √(np(1 − p))
Features
- P(X = k), ≤, <, ≥ and > k
- Mean, variance and SD
- Formula with your values
- Full distribution table and CSV
- Accurate for n up to 10,000
Frequently asked questions
What is the binomial probability formula?
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) = n! ÷ (k!(n − k)!) counts the ways to choose k successes.
What is the probability of exactly 5 heads in 10 flips?
C(10, 5) × 0.5^10 = 252 ÷ 1,024 = 0.2461, about 24.6%.
What is the difference between at most and at least?
At most k means k or fewer, P(X ≤ k). At least k means k or more, P(X ≥ k) = 1 − P(X ≤ k − 1).
When does a binomial distribution apply?
When there is a fixed number of trials, each trial has two outcomes, the trials are independent and p is the same every time.
What are the mean and standard deviation of a binomial distribution?
The mean is np and the standard deviation is √(np(1 − p)). For n = 10 and p = 0.5, the mean is 5 and the SD is 1.58.