Binomial Distribution Calculator

What Is the Binomial Distribution?

The binomial distribution models the probability of getting exactly k successes in n independent trials, where each trial has the same probability p of success (like flipping a coin, passing/failing a test, or a product being defective). It's one of the most commonly used discrete probability distributions in statistics.

How to Use This Calculator

• Enter the number of trials (n), such as the number of coin flips or attempts.
• Enter the probability of success on a single trial as a decimal between 0 and 1 (0.5 for a fair coin, 0.2 for a 20% chance, etc.).
• Enter the specific number of successes (k) you want the probability for.

The Formula Used

P(X = k) = C(n, k) × p^k × (1−p)^(n−k), where C(n, k) = n! ÷ (k! × (n−k)!) is the binomial coefficient (number of ways to choose k successes from n trials). The cumulative probability P(X ≤ k) sums this formula for every value from 0 up to k. Mean = n × p. Standard Deviation = √(n × p × (1−p)).

Worked Example

Flipping a fair coin (p = 0.5) 10 times, the probability of getting exactly 4 heads is P(X=4) = C(10,4) × 0.5⁴ × 0.5⁶ = 210 × 0.0009766 ≈ 0.205, or about 20.5%. The expected value (mean) is 10 × 0.5 = 5 heads, with a standard deviation of √(10 × 0.5 × 0.5) ≈ 1.58.

When to Use the Binomial Distribution

Use it whenever you have a fixed number of independent trials, each with only two possible outcomes (success/failure) and the same probability of success each time — common examples include quality control defect rates, A/B test conversions, and simple win/loss probability scenarios.