Binomial and Geometric Distributions (9709 Statistics 1 – 5.4)

Core Concept

The Binomial Distribution $B(n, p)$

A random variable $X$ follows a Binomial distribution when the following four conditions hold:

1. There are a fixed number of trials, $n$.
2. Each trial results in exactly one of two outcomes — success or failure.
3. Trials are independent of each other.
4. The probability of success, $p$, is constant across every trial.

$X$ counts the total number of successes in $n$ trials. We write $X \sim B(n, p)$.

Practical examples: the number of defective items in a batch of 20, the number of heads in 10 coin flips, the number of sixes in 8 rolls of a fair die.

The Geometric Distribution $\text{Geo}(p)$

A random variable $R$ follows a Geometric distribution when:

1. Trials are independent.
2. The probability of success, $p$, is constant.
3. Trials continue until the first success.

$R$ counts the trial number on which the first success occurs, so $R \in \{1, 2, 3, \ldots\}$ (there is no upper bound). We write $R \sim \text{Geo}(p)$.

Practical examples: the number of attempts needed to roll a six, the number of components tested until the first faulty one is found, the number of interviews until a candidate receives a job offer.

Key distinction: Binomial = fixed $n$, count successes. Geometric = no fixed $n$, count trials until first success.