Binomial Distribution Calculator: Easily Compute Probability Mass Function

How to use the tool

1. Fill n — trials: type 12 or 45 (whole numbers only).
2. Fill p — success probability: enter 0.40 or 0.18 (range 0–1).
3. Fill k — successes: choose any integer 0 ≤ k ≤ n, e.g., 3 or 20.
4. Press “Calculate”: the result shows P(X = k).
5. Reset fields: hit “Clear” before a new scenario.

Formulas the script applies

Exact probability mass function:

$$P(X=k)=rac{n!}{k!\,(n-k)!}\,p^{k}(1-p)^{n-k}$$

Mean: $$\mu = n p$$    Variance: $$\sigma^{2}=n p(1-p)$$

Worked examples

• Example 1: n = 12, p = 0.40, k = 3  220 × 0.4³ × 0.6⁹ = 0.1411.
• Example 2: n = 5, p = 0.70, k = 4  5 × 0.7⁴ × 0.3 = 0.3602.

Quick-Facts

• Discrete two-outcome model; trials assumed independent (NIST/SEMATECH e-Handbook, 2023).
• Mean = np, Variance = np(1−p) (Ross, 2014).
• Normal approximation reasonable when n p ≥ 5 and n(1−p) ≥ 5 (Walpole et al., 2017).
• Exact JavaScript factorial safe up to n≈170 before overflow (MDN Web Docs, 2023).

FAQ

What is the binomial distribution?

The binomial distribution counts the number of successes in n identical, independent Bernoulli trials with constant probability p (Ross, 2014).

When should I use this calculator?

Use it whenever you need an exact P(X = k) for binary outcomes, such as pass/fail tests or defect counts (NIST, 2023).

Why must n and k be integers?

Trials and successes are count data; fractional counts violate the underlying Bernoulli model (Walpole et al., 2017).

Can the tool show cumulative probability?

You can sum successive exact outputs; cumulative mode is planned but not yet implemented.

How large can n be?

Exact factorial math stays stable up to about 170; beyond that, numerical libraries or logarithms are advised (MDN, 2023).

What happens if p = 0 or 1?

Probability collapses: P(X = 0) = 1 when p = 0; P(X = n) = 1 when p = 1, all else zero.

Is a normal approximation acceptable?

Yes, if n p and n(1−p) are both at least 5; accuracy improves as these products grow (Walpole et al., 2017).

Expert tip

“The binomial model is exact; use it whenever computationally feasible,” advises the NIST e-Handbook (2023).