Binomial Distribution Calculator
Compute P(X = k), P(X ≤ k), and P(X ≥ k) for any binomial distribution B(n, p) instantly. Enter the number of trials n, probability of success p, and number of successes k to get exact probabilities, cumulative probabilities, mean, variance, and a full bar chart visualization. Free, private, and 100% client-side. No signup required.
Enter the number of trials (n), probability of success (p), and number of successes (k) to compute P(X = k), P(X ≤ k), and P(X ≥ k) with a full bar chart visualization. All calculations run locally in your browser.
Positive integer, max 1,000
Between 0 and 1
Integer, 0 ≤ k ≤ n
- n = number of independent trials
- p = probability of success on each trial (0 ≤ p ≤ 1)
- k = number of successes (0 ≤ k ≤ n)
- Mean = np, Variance = np(1−p), Std Dev = √(np(1−p))
The most complete binomial distribution calculator online — computes exact, cumulative, and survival probabilities with a full bar chart visualization.
Instant Binomial Distribution Calculation
Compute P(X = k), P(X ≤ k), and P(X ≥ k) for any B(n, p) distribution instantly. The binomial distribution calculator handles n up to 1,000 using log-space arithmetic for numerical precision.
Full Bar Chart Visualization
See the complete probability mass function as an interactive bar chart. The selected k value is highlighted in primary color, P(X ≤ k) is shaded, and the chart auto-scales for any n and p combination.
Secure Binomial Calculator Online
Every calculation runs entirely in your browser using client-side JavaScript. Your inputs never leave your device — complete privacy, zero server uploads, no tracking.
100% Free — No Signup Required
The binomial distribution calculator is completely free with no account, no ads, and no usage limits. Calculate as many distributions as you need, any time, on any device.
See how statisticians, engineers, students, and analysts use the binomial distribution calculator in their daily work.
Statistics & Probability Homework
Students use the binomial distribution calculator to verify homework answers and understand cumulative probabilities. Enter n, p, and k to instantly see P(X = k), P(X ≤ k), and P(X ≥ k) with the full PMF formula.
Quality Control & Manufacturing
Quality engineers use the binomial distribution calculator to compute the probability of finding exactly k defective items in a batch of n. For example, P(X ≤ 2) for n=50 items with a 2% defect rate.
Clinical Trials & Medical Research
Researchers use the binomial distribution calculator to compute the probability of observing k successes in n patients given a treatment success rate p. Essential for power analysis and hypothesis testing.
Finance & Risk Analysis
Analysts use the binomial distribution calculator to model binary outcomes — loan defaults, option exercises, or market up/down days. Compute the probability of k defaults in a portfolio of n loans.
A/B Testing & Conversion Rates
Product managers and data scientists use the binomial distribution calculator to evaluate A/B test results. Compute the probability of observing k conversions in n visitors given a baseline conversion rate p.
Genetics & Biology
Biologists use the binomial distribution calculator to model Mendelian inheritance. For example, the probability of exactly k offspring showing a dominant trait in n offspring given p = 0.75 for a heterozygous cross.
A complete reference guide to the binomial distribution, its parameters, formulas, and when to use it.
What is the Binomial Distribution?
The binomial distribution B(n, p) models the number of successes in n independent trials, each with the same probability of success p. It applies whenever you have a fixed number of trials, each trial has exactly two outcomes (success or failure), trials are independent, and the probability of success is constant. Classic examples include coin flips, quality control inspections, clinical trial outcomes, and A/B test conversions.
How Our Binomial Distribution Calculator Works
- 1Enter n, p, and k: n is the number of trials, p is the probability of success per trial (0–1), and k is the number of successes you want to evaluate.
- 2Click Calculate: The calculator applies the PMF formula using log-space arithmetic for numerical precision, then sums probabilities for the CDF and survival function. All calculations run locally in your browser — your inputs never leave your device.
- 3Read the results: See P(X = k), P(X ≤ k), and P(X ≥ k) as both decimals and percentages, plus the mean, variance, standard deviation, and mode. The bar chart shows the full probability mass function with your k highlighted.
What Gets Computed
- P(X = k) — Exact Probability (PMF): The probability of getting exactly k successes. Formula: C(n,k) × p^k × (1−p)^(n−k).
- P(X ≤ k) — Cumulative Probability (CDF): The probability of getting at most k successes. Computed as the sum of P(X = i) for i = 0 to k.
- P(X ≥ k) — Survival Function: The probability of getting at least k successes. Computed as 1 − P(X ≤ k−1).
- Distribution Properties: Mean = np, Variance = np(1−p), Standard Deviation = √(np(1−p)), Mode = ⌊(n+1)p⌋.
PMF vs. CDF
The PMF (P(X = k)) gives the probability of exactly k successes. The CDF (P(X ≤ k)) gives the probability of k or fewer successes — it's the sum of all PMF values from 0 to k.
Log-Space Arithmetic
For large n, computing C(n,k) directly overflows. The calculator uses log-space: log P(X=k) = log C(n,k) + k·log(p) + (n−k)·log(1−p), then exponentiates for precision.
Normal Approximation
For large n (n > 1,000), the binomial distribution is well-approximated by a normal distribution N(np, np(1−p)). This calculator supports n up to 1,000 exactly.
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A binomial distribution calculator computes probabilities for the binomial distribution B(n, p) — the number of successes in n independent trials each with probability p. Our calculator computes P(X = k), P(X ≤ k), and P(X ≥ k) with a full bar chart visualization, and runs entirely in your browser with no signup required.
P(X = k) is the probability of getting exactly k successes in n trials. It is computed using the PMF formula: P(X = k) = C(n, k) × p^k × (1−p)^(n−k), where C(n, k) is the binomial coefficient "n choose k". For example, P(X = 5) for B(10, 0.5) = C(10,5) × 0.5^5 × 0.5^5 ≈ 0.2461.
P(X ≤ k) is the cumulative distribution function (CDF) — the probability of getting at most k successes. It equals the sum of P(X = i) for i = 0 to k. Use P(X ≤ k) when you want to know the probability of "k or fewer" successes, such as "what is the probability of at most 2 defects in a batch of 50?"
P(X ≥ k) is the survival function — the probability of getting at least k successes. It equals 1 − P(X ≤ k−1). Use P(X ≥ k) when you want to know the probability of "k or more" successes, such as "what is the probability of at least 3 conversions in 20 visitors?"
The binomial distribution applies when: (1) there are a fixed number of trials n, (2) each trial has exactly two outcomes (success or failure), (3) trials are independent of each other, and (4) the probability of success p is the same for every trial. If these conditions are not met, other distributions (Poisson, hypergeometric, negative binomial) may be more appropriate.
For B(n, p): Mean = np, Variance = np(1−p), Standard Deviation = √(np(1−p)). For example, B(10, 0.5) has mean = 5, variance = 2.5, and standard deviation ≈ 1.581. The mode (most likely value) is ⌊(n+1)p⌋.
For n > 1,000, computing the exact binomial probabilities becomes very slow (the CDF requires summing up to 1,001 terms). For large n, the normal approximation N(np, np(1−p)) is highly accurate and much faster. The calculator uses log-space arithmetic to handle n up to 1,000 with full precision.
Yes, completely. All calculations are performed locally in your web browser using client-side JavaScript. No inputs, results, or parameters are ever sent to any server. Your data stays entirely on your device.
Yes. The binomial distribution calculator is 100% free with no account registration, no payment, and no usage limits. Calculate as many distributions as you need, any time, on any device.