Formula & Calculator
Binomial Probability
Probability of exactly k successes in n independent Bernoulli trials.
Interpretation
P(X=k) = C(n,k) pᵏ(1−p)ⁿ⁻ᵏ. Probability of k successes in n independent trials, each with probability p. Used in quality control, genetics, and opinion polling.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| P(X=k) | Probability of k successes | |
| n | Number of trials | |
| p | Success probability | |
| k | Number of successes |
What it means
The binomial distribution models the number of successes (k) in a fixed number n of independent trials, where each trial has the same probability p of success. The formula uses the binomial coefficient C(n,k) to count the number of ways to choose k successes, multiplied by pᵏ and (1−p)ⁿ⁻ᵏ. It is used in quality control (number of defects in a sample), in genetics (inheritance patterns), in polling (number of favourable responses), and in reliability engineering. The distribution is discrete and is the basis for many statistical tests (e.g., sign test). Understanding the binomial probability is fundamental for analyzing experiments with binary outcomes and for constructing confidence intervals for proportions.
Worked example
Binomial Probability – Two Examples
Real‑World| Parameter | Value |
|---|---|
| n | 5 |
| k | 2 |
| p | 0.5 |
| Parameter | Value |
|---|---|
| n | 15 |
| k | 5 |
| p | 0.30 |
Common mistakes
- n and k: k must be an integer between 0 and n – inclusive.
- Probability p: The probability of success on a single trial – must be between 0 and 1.
- Combination C(n,k): Number of ways to choose k successes – compute correctly.
- Independence: Assumes independent trials with constant p.
- Expected value: E(X) = np – not n*p*(1-p) (that’s variance).
Applications
The binomial probability formula gives the probability of exactly k successes in n independent trials, each with probability p of success. This distribution models countless binary outcome scenarios in engineering, science, and business. Engineers use it in quality control (probability of defective items), in reliability (number of failures), and in survey sampling. In biology, it models genetic traits. In finance, it helps predict the number of defaults. By calculating binomial probabilities, professionals can quantify the likelihood of various outcomes, set acceptance criteria, and design experiments. The binomial distribution is the basis for many hypothesis tests and confidence intervals for proportions. Understanding this formula is essential for any work involving counts of successes in repeated trials.
- Quality control – acceptance sampling and defect rates
- Reliability engineering – number of failures in a system
- Survey and polling analysis – estimating proportions
- Genetics – inheritance patterns and trait frequencies
- Finance – probability of defaults or successes in portfolios
Frequently Asked Questions
P(X=k) = C(n,k) pᵏ (1−p)ⁿ⁻ᵏ. It gives the probability of exactly k successes in n independent Bernoulli trials, each with success probability p.
- Fixed number of trials n.
- Each trial is independent.
- Only two outcomes: success/failure.
- Constant probability p for each trial.
E[X] = np and Var(X) = np(1−p). These are derived from the properties of Bernoulli trials.
For large n and not too extreme p, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1−p) (Central Limit Theorem). This is the basis for the normal approximation to the binomial.
A Bernoulli trial is a single experiment (n=1). The binomial distribution is the sum of n independent Bernoulli trials, representing the total number of successes.
Sum the probabilities from k to n: P(X ≥ k) = Σ_{i=k}^{n} C(n,i) pⁱ (1−p)ⁿ⁻ⁱ. This is often computed using cumulative distribution functions or tables.
If success probability is p, the failure probability is q = 1−p. The number of failures follows a binomial distribution with parameters n and q.
- Quality control: number of defective items in a batch.
- Polling: number of favourable responses.
- Genetics: number of offspring with a certain trait.
- Medicine: number of patients who recover from a treatment.
C(n,k) = n! / (k!(n−k)!). It counts the number of ways to choose k successes out of n trials, regardless of order.
When using the normal approximation, adjust the boundaries by ±0.5 (e.g., P(X ≤ k) becomes P(X ≤ k+0.5)). This improves accuracy for discrete‑to‑continuous approximation.