Module 3 · Distributions · Topic 4
Binomial Setting. A binomial experiment has a fixed number of trials (n), each trial has exactly two outcomes (success/failure), trials are independent, and the probability of success (p) is constant.
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
μ = np, σ² = np(1 − p)
P(X = k) = C(n,k)p^k(1-p)^(n-k) — Exact probability of k successes in n independent Bernoulli trials.z = (x - mu) / sigma — Standardizes a value by measuring standard deviations from the mean.A quiz has 5 true/false questions answered randomly. n = 5, p = 0.5. P(exactly 3 correct) = C(5,3) × 0.5³ × 0.5² = 10 × 0.03125 = 0.3125.
Use the binomial when you have a fixed number of independent yes/no trials with the same success probability. Common examples: coin flips, pass/fail quality checks, multiple-choice guessing.
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