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Enter the number of trials, success probability, and target successes to see the binomial probabilities.
What Is a Binomial Distribution?
A binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success. Coin flips, quality-control checks, survey responses, and medical test results are all classic examples. The calculator finds the probability of an exact number of successes, the probability of that many successes or fewer, and the probability of that many successes or more.
The model only applies when the trials are independent, the number of trials is fixed, there are exactly two outcomes per trial, and the success probability stays the same every time. If any of those assumptions fail, the binomial formula will not match reality.
Binomial Probability Formula
- n = number of trials
- k = target number of successes
- p = probability of success on a single trial
- C(n, k) = "n choose k", the number of ways to pick k successes from n trials
When to Use This Calculator
Use the calculator whenever you want to know how likely a count of successes is. For example: "If I flip a fair coin 10 times, what is the chance of getting exactly 6 heads?" or "If 3% of items are defective, what is the chance of finding at least 2 defectives in a sample of 50?" The tool also returns the cumulative probabilities P(X ≤ k) and P(X ≥ k), which are often the quantities people actually need.
Mean, Variance, and Standard Deviation
For a binomial distribution, the expected number of successes is simply n times p. The variance measures how spread out the outcomes are around that average, and the standard deviation is the square root of the variance. These values help you decide whether an observed count is close to typical or unusually far from the mean.
| Measure | Formula |
|---|---|
| Expected value (mean) | μ = n × p |
| Variance | σ² = n × p × (1 − p) |
| Standard deviation | σ = √(n × p × (1 − p)) |
Worked Example
Suppose a basketball player makes 70% of free throws and attempts 8 in a game. What is the probability of making exactly 6?
- n = 8, p = 0.70, k = 6
- C(8, 6) = 28
- P(X = 6) = 28 × 0.70⁶ × 0.30² ≈ 0.2965, or about 29.65%
The calculator also reports that the chance of 6 or fewer makes is about 74.4% and the chance of 6 or more is about 55.1%. The expected number of made free throws is 8 × 0.70 = 5.6.
Binomial Distribution FAQ
What does P(X = k) mean? +
It is the probability that the number of successes X is exactly equal to k.
What is the difference between P(X ≤ k) and P(X ≥ k)? +
P(X ≤ k) is the cumulative probability of k successes or fewer. P(X ≥ k) is the probability of k successes or more. They are useful for one-sided questions such as 'at most' or 'at least'.
Can p be entered as a percentage? +
Yes. Enter 0.70 or 70; the calculator converts percentages to decimals automatically.
What if p is 0 or 1? +
If p is 0, success is impossible so P(X = 0) = 1. If p is 1, success is guaranteed so P(X = n) = 1. The calculator handles these edge cases.
How large can n be? +
The calculator accepts up to 10,000 trials. Larger values become expensive to compute and are usually approximated by a normal distribution.
When is the binomial model not appropriate? +
When trials are not independent, the success probability changes between trials, or there are more than two possible outcomes. In those cases use a different probability model.
External Resources
Authoritative references for binomial probability and statistics.
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