Geometric Distribution — Definition, Formula & Examples

The geometric distribution models the number of independent Bernoulli trials needed to get the first success. It answers the question: how many attempts will it take before something happens for the first time?

A discrete probability distribution where the random variable $X$ represents the trial number of the first success in a sequence of independent Bernoulli trials, each with constant probability of success $p$ . The support is $X \in \{1, 2, 3, \dots\}$ , and the distribution is memoryless: $P(X > m + n \mid X > m) = P(X > n)$ .

Key Formula

P(X = k) = (1 - p)^{k-1} \cdot p

Where:

$k$ = Trial number on which the first success occurs (k = 1, 2, 3, ...)
$p$ = Probability of success on each trial (0 < p ≤ 1)

How It Works

Each trial has exactly two outcomes—success (probability

p

) or failure (probability

1-p

)—and trials are independent. To find the probability that the first success occurs on trial

k

, you need exactly

k-1

consecutive failures followed by one success. The expected number of trials until the first success is

\frac{1}{p}

, so rarer events require more attempts on average. The variance is

\frac{1-p}{p^2}

Worked Example

Problem: A free-throw shooter makes 80% of her shots. What is the probability that her first miss occurs on the 4th shot?

Identify parameters: Here 'success' is a miss, so p = 0.20 (miss probability) and we want k = 4.

p = 0.20, \quad k = 4

Apply the formula: She must make 3 shots in a row (each with probability 0.80) then miss on the 4th.

P(X = 4) = (1 - 0.20)^{4-1} \cdot 0.20 = (0.80)^3 \cdot 0.20

Calculate: Compute the result.

P(X = 4) = 0.512 \cdot 0.20 = 0.1024

Answer: The probability that her first miss occurs on the 4th shot is 0.1024, or about 10.2%.

Visualization

Why It Matters

The geometric distribution appears on the AP Statistics exam and in college probability courses. It is used in quality control (how many items until a defect?), network engineering (how many packets until a lost one?), and any scenario where you wait for a first occurrence.

Common Mistakes

Mistake: Confusing the geometric distribution with the binomial distribution.

Correction: The binomial counts total successes in a fixed number of trials. The geometric counts trials until the first success—the number of trials is random, not fixed.