The binomial distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success, like the number of heads in 10 coin tosses.

Key facts

  • X ~ B(n, p): n trials, probability p of success each time.
  • P(X = r) = nCr pʳ(1 − p)ⁿ⁻ʳ.
  • Mean np; variance np(1 − p).
  • Use cumulative tables or a calculator for P(X ≤ r).

Worked examples

Example 1

X ~ B(10, 0.3). Find P(X = 2).

  1. 10C2 × 0.3² × 0.7⁸
  2. 45 × 0.09 × 0.05765 = 0.2335

Answer: 0.2335

Example 2

A dice is rolled 30 times. Find the mean and variance of the number of sixes.

  1. 30 × 1/6 = 5
  2. 30 × 1/6 × 5/6 ≈ 4.17

Answer: Mean 5, variance 4.17