Math & Business Productivity

Binomial Distribution & Probability Calculator

Free binomial distribution calculator. Compute exact probability P(X=k), cumulative P(X<=k), at-least P(X>=k), mean, variance, and standard deviation.

Reviewed by Dr. Marcus Vance, Ph.D. (Ph.D. Applied Mathematics (MIT))

Verified for accuracy•Updated 2026

Interactive Calculator

1. Provide Details

Number of Trials (n)

trials

Probability of Success (p, 0 to 1)

rate

Number of Successes (k)

successes

2. Output Results

Exact Probability P(X = k)24.6094% (0.246094)

Cumulative Probability P(X ≤ k)62.3047% (0.623047)

At Least Probability P(X ≥ k)62.3047% (0.623047)

Expected Value / Mean (μ)5.0000

Standard Deviation (σ)1.5811 (Var: 2.5000)

How to Calculate: Formula & Steps

Calculates the exact binomial probability of getting k successes in n independent Bernoulli trials with constant success rate p.

Formula Used:P(X = k) = C(n, k) × p^k × (1 - p)^(n - k)

Step-by-Step Calculation Example

For n = 10 trials, p = 0.5 success rate, and k = 5 successes: P(X = 5) = C(10,5) × 0.5⁵ × 0.5⁵ = 252 × 0.00097656 = 0.2461 (24.61%).

Common Mistakes to Avoid

Entering success probability p as a percentage (e.g. 50) instead of a decimal between 0 and 1 (0.50).
Confusing exact probability P(X = k) with cumulative probability P(X ≤ k).

Practical Use Cases

A/B conversion test trial confidence analysis
Manufacturing quality control defect sampling
Coin flip and sports probability forecasting

Expert Tips

Verify that trials (n) are strictly independent and that probability (p) stays constant throughout.

Disclaimer

This calculation tool is provided for educational and informational estimation purposes only. Results are based on mathematical formulas and user-supplied parameters. They do not constitute formal underwriting, financial, tax, engineering, or legal determinations.

Frequently Asked Questions About Binomial Distribution & Probability Calculator

What is a binomial distribution?

A binomial distribution describes the discrete probability distribution of obtaining exactly k successes in n independent Bernoulli trials, where each trial has a constant success probability p.

How do I calculate exact probability P(X = k)?

Use the binomial PMF formula: P(X = k) = [n! / (k! × (n - k)!)] × p^k × (1 - p)^(n - k). Use our free Binomial Distribution Calculator to get instant results.

What is the difference between PMF and CDF in binomial distribution?

PMF (Probability Mass Function) calculates the exact chance of obtaining exactly k successes. CDF (Cumulative Distribution Function) sums probabilities to find the chance of getting up to k successes, P(X ≤ k).

How are mean and variance calculated for a binomial distribution?

The mean (expected value) is μ = n × p. The variance is σ² = n × p × (1 - p), and the standard deviation is σ = √(n × p × (1 - p)).