2.12 The Geometric Distribution

What makes a geometric setting?

A geometric setting requires: (1) each trial has only two outcomes — success or failure, (2) the probability of success p is the same every trial, (3) trials are independent, and (4) the variable of interest is the number of trials until the first success. Unlike binomial, there's no fixed number of trials.

Geometric probability formula

If X = trial number of the first success, then P(X = k) = (1-p)^(k-1) · p, for k = 1, 2, 3, .... This means you need (k-1) failures followed by one success. The distribution is skewed right — small values of k are most likely.

Mean and standard deviation

For a geometric random variable: mean (expected value) = 1/p and standard deviation = √((1-p)/p²). A smaller p means it takes longer on average to get a success, so the mean increases as p decreases.

Key terms

Geometric setting conditions

Two outcomes per trial, constant probability of success, independent trials, and variable = trials until first success.

Geometric probability formula

P(X = k) = (1-p)^(k-1) · p, where k is the trial number of the first success.

Mean of geometric distribution

1/p — the expected number of trials until the first success.

Standard deviation of geometric distribution

√((1-p)/p²)

Shape of geometric distribution

Skewed right; probability is highest at X = 1 and decreases as X increases.

Try an AP-style question

p = 0.2 per trial. What is P(first success on trial 3)?

1. 0.2
2. 0.488
3. 0.128
4. 0.008