2.10 Introduction to the Binomial Distribution

What makes a Binomial Setting?

A binomial setting requires: (1) each trial has only two outcomes (success/failure), (2) a fixed number of trials n, (3) trials are independent, and (4) the probability of success p is constant across trials. Remember with BINS: Binary, Independent, Number fixed, Same probability.

The Binomial Random Variable

If X counts the number of successes in a binomial setting, X is a binomial random variable with parameters n and p, written X ~ B(n, p). Its possible values are 0, 1, 2, ..., n. The mean is μₓ = np and the standard deviation is σₓ = √(np(1−p)).

Binomial Probability Formula

P(X = k) = (n choose k) · pᵏ · (1−p)ⁿ⁻ᵏ, where (n choose k) = n! / (k!(n−k)!) counts the number of arrangements of k successes among n trials. This gives the probability of exactly k successes in n independent trials.

Key terms

Binomial setting (BINS)

Binary outcomes, Independent trials, fixed Number of trials n, Same probability p of success each trial.

Binomial random variable

X ~ B(n, p): counts the number of successes in n independent trials, each with success probability p.

Mean of binomial distribution

μₓ = np

Standard deviation of binomial distribution

σₓ = √(np(1−p))

Binomial probability formula

P(X = k) = (n choose k) · pᵏ · (1−p)ⁿ⁻ᵏ, where (n choose k) = n!/(k!(n−k)!)

Try an AP-style question

A fair coin is flipped 4 times. What is P(exactly 2 heads)?

1. 0.25
2. 0.375
3. 0.5
4. 0.0625