Binomial Distribution Calculator

Frequently Asked Questions

When is the binomial distribution used?

For a fixed number n of independent yes/no trials each with success probability p - counting total successes (coin flips, defect rates, conversion counts).

What does the calculator output?

P(X=k) and the cumulative probabilities (≤, <, ≥, >), plus mean np, variance np(1−p), and standard deviation.

When can I approximate it with a normal distribution?

When np and n(1−p) are both ≳ 10, the binomial is approximately normal with the same mean and variance (use a continuity correction).

What is the difference from the Poisson distribution?

Binomial has a fixed n and probability p. Poisson models counts over an interval with rate λ and no fixed n - it is the limit of the binomial as n→∞, p→0.

Provided by AllCalculators.io
Free online calculators for everyday. No registration required.

Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

JavaScript is required to use the interactive calculator above. The questions and answers below remain readable without JavaScript.

How This Calculator Works

The binomial distribution models the number of successes in n independent trials, each with the same success probability p. The probability of exactly k successes is the binomial PMF: P(X = k) = C(n,k) p^k (1 − p)^n−k, where C(n,k) = n! ÷ [k!(n − k)!] is the binomial coefficient. To stay numerically stable for large n this calculator works in log space - summing log-factorials rather than multiplying huge factorials directly - then exponentiates. Cumulative results are exact finite sums: P(X ≤ k) = Σ_i=0..k P(X = i), with P(X < k), P(X ≥ k), and P(X > k) derived by complement. The mean is np, the variance is np(1 − p), and the standard deviation is √[np(1 − p)].

Worked Example: n = 10, p = 0.5, k = 5

Ten fair coin flips, probability of exactly 5 heads. C(10,5) = 252, p⁵(1−p)⁵ = 0.5¹⁰ = 1 ÷ 1024 ≈ 0.0009766. So P(X = 5) = 252 × 0.0009766 ≈ 0.2461 - the single most likely outcome, yet still under 25%. The mean is np = 5, the variance is np(1 − p) = 2.5, and the standard deviation is √2.5 ≈ 1.581. The cumulative P(X ≤ 5) ≈ 0.6230 by symmetry of the fair-coin case.

Common Mistakes & Assumptions

The four binomial conditions must hold: a fixed number of trials n, only two outcomes per trial, a constant p across trials, and independence between trials. Sampling without replacement from a small population violates the constant-p and independence assumptions - use the hypergeometric distribution there. Confusing "at least k" with "more than k" is a frequent slip: P(X ≥ k) includes k itself, while P(X > k) excludes it. Probabilities must be entered as decimals (0.05, not 5).

Interpretation & Limitations

P(X = k) for any single k is often small even at the mode - that is normal, since probability is spread across n + 1 possible counts. For large n with p near 0.5 the binomial is well approximated by a normal with the same mean and variance (apply a continuity correction); for large n with small p it approaches a Poisson with λ = np. This tool computes the exact binomial, so it remains correct in those regimes but factorial summation for very large n (tens of thousands) becomes slow. Results are accurate to the displayed six decimals.

Binomial Distribution Calculator

Frequently Asked Questions

Related Statistics Calculators