The Geometric Distribution

Waiting for Success

The Geometric Distribution answers a fundamental question: How many trials do I need to perform before I see my first success?

Unlike the Binomial distribution, where the number of trials ( $n$ ) is fixed and the number of successes is random, the Geometric distribution fixes the number of successes (exactly 1) and makes the number of trials random.

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

Here, $X$ is the total number of trials. The formula is intuitive: to have the first success on the $k$ -th trial, you must have exactly $k-1$ consecutive failures (probability $(1-p)^{k-1}$ ) followed by a success (probability $p$ ).

Core Properties

Mean ( $E[X]$ ): $\frac{1}{p}$
Variance ( $Var(X)$ ): $\frac{1-p}{p^2}$
Support: $k \in \{1, 2, 3, \dots\}$

EExample

Real-World Examples

Manufacturing: Testing items coming off an assembly line until a defective one is found.
Sales: Calling potential clients until one agrees to a meeting.
Gaming: Grinding for a rare item drop with a 1% chance in a video game ( $p=0.01$ ). You expect to wait $1 / 0.01 = 100$ tries on average.

The Memoryless Property

The Geometric distribution is the only discrete probability distribution that exhibits the memoryless property. This means that the probability of success on the next trial is independent of how many failures have occurred so far.

P(X > a + b \mid X > a) = P(X > b)

If you've flipped a coin 10 times and haven't gotten heads, the probability that it takes 5 more flips to get heads is exactly the same as if you were starting fresh. The coin has no memory of the previous 10 failures.

✦Intuition

Why is the Mean 1/p?

If a game has a 1-in-5 chance of winning ( $p=0.2$ ), intuition tells us we should play about 5 times to win once. Mathematically, the expected value of the Geometric distribution is exactly $1/p$ . While you could win on the very first try or be unlucky and take 20 tries, across infinite parallel universes, the average number of attempts will converge tightly to $1/p$ .

Advanced Practice

Example 1: The Gacha Roll

medium

In a video game, opening a 'loot box' has a 2% chance of yielding a rare item ( $p=0.02$ ). What is the probability that you get your first rare item on exactly your 50th attempt?

Example 2: Surviving the Filter

hard

A telemarketer has a 10% chance of making a sale on any given call. What is the probability that they need to make more than 3 calls to get their first sale?

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The Binomial Distribution

Course Progression

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The Hypergeometric Distribution