Binomial Distribution Calculator

Binomial Distribution Guide

Binomial Distribution Conditions

The binomial distribution B(n, p) applies when: fixed number of trials n; each trial is independent; only two outcomes (success/failure); constant probability of success p. P(X = r) = C(n,r) × p^r × (1-p)^(n-r). Where C(n,r) = n!/(r!(n-r)!) = binomial coefficient. Mean μ = np. Variance σ² = np(1-p). Standard deviation σ = √(np(1-p)). Examples: number of heads in 20 coin flips, number of defective items in a batch, number of patients responding to treatment.

Cumulative Binomial Probabilities

P(X ≤ r) = Σ P(X = k) for k = 0 to r. This is the cumulative distribution function (CDF). P(X ≥ r) = 1 − P(X ≤ r-1). P(X > r) = 1 − P(X ≤ r). In practice, statistical tables give cumulative binomial probabilities — the tables typically show P(X ≤ r) for various n, p, and r combinations. Calculator approach: sum individual probabilities P(X=0) + P(X=1) + ... + P(X=r). Most scientific calculators have a built-in binomial CDF function.

Normal Approximation to Binomial

When n is large and p is not close to 0 or 1, the binomial distribution approximates a normal distribution N(np, np(1-p)). Rule of thumb: use the normal approximation when np ≥ 5 AND n(1-p) ≥ 5. Continuity correction: P(X ≤ r) in binomial ≈ P(X ≤ r + 0.5) in normal. The correction accounts for the fact that normal is continuous and binomial is discrete. For p close to 0.5 and large n, the approximation is very accurate. For p close to 0 or 1, a Poisson distribution may be a better approximation.

Real-World Applications

Quality control: if a manufacturing process produces 2% defective items (p=0.02), what is the probability that a sample of 100 items contains more than 5 defectives? Genetics: in a dominant/recessive cross (Aa × Aa), each offspring has a 3/4 probability of showing the dominant phenotype. In a family of 6 children, P(exactly 4 dominant) = C(6,4) × (3/4)⁴ × (1/4)² = 0.297. Medical trials: a drug has 70% success rate. In 15 patients, P(at least 12 succeed) = P(X≥12) = 1 − P(X≤11). Sports: a basketb