Enter the number of trials (n), probability of success (p), and number of successes (k) to compute exact and cumulative binomial probabilities.
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It's the foundation of A/B testing, quality control, polling, clinical trials, and any scenario where you have a series of yes/no outcomes. The classic example: flip a coin 10 times — what's the probability of getting exactly 7 heads? The binomial distribution gives the answer: 0.117 (about 12%).
This calculator returns the exact probability P(X = k), cumulative probability P(X ≤ k), and tail probability P(X ≥ k) for binomial distributions. Use it for homework problems, hypothesis testing, sample size planning, or quick reality checks on probabilities.
The binomial requires three things: a fixed number of trials (n), independence between trials, and constant probability of success (p) across trials. The mean is np, variance is np(1-p), and the distribution is approximately normal when n is large and p isn't extreme.
**Scenario:** Test version converts 10% of visitors. Run experiment with 50 visitors. What's P(8 or more conversions)? **Calculation:** n=50, p=0.10, k=8. Mean = 5; SD = 2.12. P(X≥8) using binomial: ~0.092. **Result:** ~9% probability of seeing 8+ conversions. Statistical significance threshold (α=0.05) requires more extreme; current result is borderline.
**Scenario:** Factory produces 5% defective parts. Sample 20 parts. P(2 or fewer defective)? **Calculation:** n=20, p=0.05, k=2. P(X≤2) = P(0)+P(1)+P(2) = 0.358+0.377+0.189 = 0.924. **Result:** 92.4% probability of finding ≤2 defective parts. This is consistent with target. Sample of 20 supports the 5% defect rate hypothesis.
**Scenario:** Player makes 80% of free throws. In 10 attempts, what's P(making exactly 7)? **Calculation:** n=10, p=0.80, k=7. C(10,7) × 0.8^7 × 0.2^3 = 120 × 0.2097 × 0.008 = 0.201. **Result:** 20.1% chance of making exactly 7. Mean expected: 8. The actual outcome of 7 is plausible (within 1 SD of mean).
**Use binomial distribution for:**
- **Sample surveys**: number agreeing/disagreeing. - **Quality control**: defects in batch. - **A/B testing**: conversion rates. - **Sports statistics**: success rates per attempt. - **Genetics**: inheritance probabilities. - **Clinical trials**: response rates. - **Reliability**: component failures.
**Key checks:**
- Fixed n trials. - Each trial independent. - Probability p constant across trials. - Two outcomes per trial.
**Normal approximation when:**
- np ≥ 5 AND n(1-p) ≥ 5. - Use mean = np, variance = np(1-p). - Apply continuity correction (±0.5).
**Software:**
- **Excel**: BINOM.DIST(k, n, p, FALSE) for exact; TRUE for cumulative. - **R**: dbinom(k, n, p), pbinom(k, n, p). - **Python (scipy.stats)**: binom.pmf(k, n, p), binom.cdf(k, n, p).
**Related distributions:**
| Distribution | Difference | |---|---| | Bernoulli | n = 1 (single trial) | | Geometric | Trials until first success | | Negative binomial | Trials until r successes | | Hypergeometric | Sampling without replacement | | Poisson | Limit of binomial as n→∞, np = λ |
**Common issues:**
- Confusing with Poisson (Poisson for rate over interval). - Using when trials are dependent. - Wrong probability per trial. - Forgetting normal approximation conditions.
Calculate P(X=k) and cumulative probabilities for a Poisson distribution.
Calculate z-score, probability, and percentile for a normal distribution.
Calculate basic probability from favorable outcomes divided by total outcomes.
Calculate the interquartile range (IQR) from a data set of up to 10 values.
Find the percentile rank of a value in a data set.
Perform a z-test for hypothesis testing with a decision at your chosen significance level.
P(X = k)
0.246094
P(X <= k)
0.623047
P(X > k)
0.376953
Mean
5.0000
Variance
2.5000
Std Dev
1.5811