Binomial Probability Calculator — Calculate P(X=k)
The binomial distribution models the number of successes in n independent trials, each with probability p of success: P(X=k) = C(n,k) · p^k · (1-p)^(n-k). It's one of the most widely used discrete distributions in statistics, quality control, and biological sciences.
When to use this calculator
- Calculate the probability of exactly k successes in n independent trials.
- Solve quality control and defect rate problems.
- Evaluate probabilities in games, sports, and betting scenarios.
- Model clinical trials and pass/fail experiments.
- Solve probability and statistics course problems.
Real example: 10 coin flips, 7 heads?
- Given: n=10 flips, k=7 heads, p=0.5 (fair coin).
- Calculate C(10,7): 10! / (7! · 3!) = 120.
- Apply formula: P(X=7) = 120 × 0.5^7 × 0.5^3 = 120 × 0.0078125 × 0.125.
- Result: P(X=7) ≈ 0.1172 or 11.72%.
- Mean: μ = 10 × 0.5 = 5 heads; σ = √(10 × 0.5 × 0.5) = 1.58.
How it works
1 min readWhat is the binomial distribution?
The binomial distribution models the number of successes in n independent trials, each with the same probability p of success. It's one of the most important discrete distributions in statistics, widely used in quality control, A/B testing, and biological sciences.
Formula
P(X = k) = C(n, k) · p^k · (1-p)^(n-k)
Where:
Statistics of the binomial distribution
| Statistic | Formula |
|---|---|
| Mean (expected value) | μ = n · p |
| Variance | σ² = n · p · (1-p) |
| Standard deviation | σ = √(n · p · (1-p)) |
| Mode | floor((n+1)p) |
Common examples
When to use and common mistakes
Frequently asked questions
When should I use the binomial probability distribution?
Use binomial when you have n independent trials, each with exactly two outcomes (success/failure), and the probability of success is the same for every trial.
What does C(n,k) mean in the formula?
The binomial coefficient = n! / (k!(n-k)!). It counts the number of ways to choose k items from n items. For example, C(10,7) = 120.
What is the mean of a binomial distribution?
μ = n × p. For 100 fair coin flips, you expect an average of 50 heads. This is the long-term average outcome.
How do I calculate the standard deviation?
σ = √(n × p × (1-p)). For 100 fair coin flips: σ = √(100 × 0.5 × 0.5) = 5. This measures how spread out the results are.
Can I use this calculator for large sample sizes?
Yes, but for n > 1000 you may get rounding errors. For large n and p not extreme, use the normal approximation instead (works well when np > 5 and n(1-p) > 5).
How is the binomial distribution different from the Poisson distribution?
Binomial has a fixed n (number of trials). Poisson models rare events without a fixed n. When n is large and p is small, binomial approximates Poisson.
How do I find P(X ≥ k) or P(X > k)?
Use the complement: P(X ≥ k) = 1 - P(X ≤ k-1) and P(X > k) = 1 - P(X ≤ k). The calculator gives you P(X ≤ k) so you can compute these.
What are real-world examples of binomial probability?
Quality control (defective items), drug trials (patients recovering), manufacturing (pass/fail tests), sports (winning games), and A/B testing in marketing.