Binomial Distribution Calculator

Calculate exact binomial probabilities and normal approximations. Essential for statistics students, researchers, and quality control analysts.

Number of trials (n)

Success probability (p)

Successes (k)

Results

Enter values to calculate

Cumulative probability (P(X ≤ k))

Probability Distribution Plot

Understanding Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It's widely used in quality control, genetics, and survey sampling.

Calculation Example

For n=10 trials with p=0.5 probability:

P(X=5) = 0.2461 (exact)
P(X≤5) = 0.6230 (exact)
Normal approximation: μ=5, σ=1.581

Reference Table

n	p	Common Uses
5-20	0.1-0.9	Quality testing
20-100	0.01-0.05	Rare event analysis

FAQ

When is normal approximation appropriate?

When np ≥ 5 and n(1-p) ≥ 5. For n=10, p=0.3: np=3 (not appropriate).

How are cumulative probabilities calculated?

By summing individual probabilities from k=0 to current k value.

What are the distribution limits?

Maximum n=1000 to prevent browser performance issues.

Disclaimer: For educational purposes only. Results may vary in real-world applications due to distribution assumptions.

References

Binomial probability formula · Normal approximation theorem · Continuity correction

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Binomial Distribution Calculator - Exact and Approximate

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