About this calculator
Binomial Distribution Calculator is a professional probability and statistics tool used to calculate the probability, expectation and variance of the binomial distribution. The binomial distribution describes the probability distribution of k successes in n independent Bernoulli trials. For example, if you toss a coin 10 times, the probability of getting heads 5 times. The binomial distribution is one of the most important discrete probability distributions and is widely used in quality control, medical experiments, market research and other fields. This calculator supports calculation of single-point probability, cumulative probability, expectation, variance, standard deviation and other statistics, and provides intuitive probability distribution charts.
What it calculates
The binomial distribution calculator finds the probability of k successes in n independent trials with the same success probability.
Formula
P(X = k) = C(n,k) p^k (1-p)^(n-k).
Inputs
- Number of trials n.
- Number of successes k.
- Success probability p, from 0 to 1.
Example
| n | k | p | Meaning |
|---|---|---|---|
| 10 | 3 | 0.5 | 3 successes in 10 trials |
| 20 | 5 | 0.2 | Low success probability |
| 5 | 5 | 0.8 | All successes |
How to interpret the result
The result is the probability of exactly k successes. Cumulative probabilities can answer at most, at least, or interval questions.
Common mistakes
- Trials should be independent.
- Success probability should stay constant.
- k cannot be greater than n.
How to use
Use the binomial distribution calculator:
1. Enter the number of tests n (positive integer) 2. Enter the success probability p (0≤p≤1) 3. Select the calculation type: • P(X=k): succeed exactly k times • P(X≤k): At most k times of success • P(X≥k): succeed at least k times • P(a≤X≤b): The number of successes is within the interval 4. Enter the number of successes k 5. Click the "Calculate" button 6. View results and distribution plots
Main features
• Various probabilities: point probability, cumulative probability, interval probability • Statistics: expectation np, variance np(1-p), standard deviation • Distribution plots: histograms and cumulative distribution plots • Normal approximation: Normal approximation when n is large • Formula display: Display the binomial distribution formula • Batch calculation: calculate the probability of multiple k values • Parametric analysis: analyze the influence of n and p on the distribution • Totally free: unlimited use
Use cases
• Quality control: Sampling inspection pass rate • Medical trials: drug effectiveness analysis • Market research: consumer preference statistics • Exam Analysis: Score Probability for Multiple Choice Questions • Reliability engineering: system reliability calculations • Genetics: Genotype probability calculations • Sports Statistics: Hit Percentage Analysis • Probability Teaching: Explaining the Binomial Distribution