Answer first: Use this Binomial Distribution Calculator to compute the probability of exactly k successes (or at most/at least k) in n independent trials.
Enter the number of trials n, the probability of success on each trial p, and the target success count k. Choose whether you want P(X = k), P(X k), P(X k), or P(X k), and the calculator returns the result.
What the binomial distribution models
The binomial distribution describes the number of successes X in n independent trials when each trial has the same probability of success p. Each trial has only two outcomes: success or failure.
- n = number of trials (fixed)
- p = probability of success on one trial (0 to 1)
- X = total number of successes
- k = the success count you want the probability for
Core formula (exact probability)
The probability of getting exactly k successes is:
P(X = k) = C(n, k) 7^k (1 – p)^(n-k)
Where:
- C(n, k) is the binomial coefficient: n! / (k!(n-k)!)
- p^k accounts for k successes
- (1-p)^(n-k) accounts for the remaining failures
Common cumulative probabilities
Often you need a range probability, not just a single value. The calculator supports these options:
- P(X k) (at most k successes): sum from i = 0 to k of P(X = i)
- P(X k) (at least k successes): sum from i = k to n of P(X = i)
- P(X k) (less than k successes): sum from i = 0 to k-1 of P(X = i)
- P(X = k) (exactly k successes): use the exact formula
How to use the Binomial Distribution Calculator
Follow these steps to get a correct probability quickly:
- Set n to the number of independent trials.
- Set p to the probability of success for each trial (use a decimal like 0.2).
- Set k to the success count you want.
- Choose the probability type: exactly, at most, at least, or less than.
- Read the result. If inputs are invalid, the calculator shows what to fix.
Worked example 1: Quality control
A factory inspects n = 10 items. Each item has a p = 0.03 chance of being defective. What is the probability of getting k = 1 defective item?
You would compute P(X = 1) using the binomial formula with n=10, p=0.03, and k=1. The calculator returns the exact probability as a decimal (and you can interpret it as a percent).
Worked example 2: Click-through rates
Suppose an ad campaign shows the same message to n = 50 users. If the click-through probability is p = 0.12, what is the probability that fewer than k = 8 users click?
Select the probability type less than k and enter n=50, p=0.12, k=8. The calculator sums probabilities from 0 through 7 to give P(X < 8).
Important assumptions (so you dont get wrong answers)
Binomial probabilities rely on specific conditions:
- Independence: one trials outcome doesnt affect another trials outcome.
- Constant probability: the success rate p stays the same across all n trials.
- Fixed number of trials: you decide n in advance.
- Two outcomes only: success/failure (not multiple categories).
If these assumptions dont hold (for example, probability changes over time), you may need a different model.
Frequently Asked Questions
What is a binomial distribution, in plain language?
A binomial distribution counts how many successes happen in a fixed number of independent trials. Each trial has the same success probability p and only two outcomes. The random variable X equals the total number of successes, from 0 up to n.
How do I choose between P(X = k), P(X k), and P(X k)?
Use P(X = k) when you need exactly k successes. Use P(X k) for at most k successes (0 through k). Use P(X k) for at least k successes (k through n). The calculator handles the required sums.
What constraints must n, p, and k satisfy?
n must be a non-negative whole number (typically n 1). p must be between 0 and 1 inclusive. k must be an integer from 0 to n. If you enter values outside these ranges, the calculator flags the error.
Does the calculator output a probability or a percent?
The calculator returns the probability as a decimal between 0 and 1. You can convert it to a percent by multiplying by 100. For example, 0.25 means 25% chance. The probability type you chose determines the value.
When should I use a binomial model instead of a normal approximation?
Use the exact binomial model when n is not large or when you need accurate tail values. Normal approximations can be reasonable when n is large and p is not extremely close to 0 or 1. This calculator computes exact binomial probabilities directly.
Next steps
Try different k values to see how the probability shifts. If youre comparing scenarios, keep p and n consistent and vary only k, or keep k consistent and change n. Thats the fastest way to build intuition about binomial outcomes.