What is binomial distribution?

Binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Examples: coin flips, pass/fail tests.

What are the requirements?

Fixed number of trials (n), only two outcomes (success/failure), constant probability (p) for each trial, and trials are independent.

What does P(X = k) mean?

P(X = k) is the probability of getting exactly k successes out of n trials. For example, exactly 3 heads in 10 coin flips.

What is cumulative probability?

P(X ≤ k) gives the probability of k or fewer successes. P(X ≥ k) = 1 - P(X < k) gives probability of k or more successes.

When does binomial approximate normal?

When n is large and p is not too close to 0 or 1 (rule of thumb: np ≥ 5 and n(1-p) ≥ 5), the binomial distribution approximates a normal distribution.

Binomial Distribution Calculator

Calculate probabilities for success/failure experiments.

Parameters

Number of Trials (n)

Total number of independent trials

Probability of Success (p): 50%

Number of Successes (k)

Target number of successes

Probability Distribution

P(X = 5)

0 %

Probability of exactly 5 successes in 10 trials

P(X = 5)

Exactly 5

P(X ≤ 5)

5 or fewer

P(X ≥ 5)

5 or more

Mean (μ)

Expected successes

Variance (σ²)

Std Dev (σ)

Documentation