Modeling With the Geometric Distribution

Apply the geometric distribution to real-world scenarios: compute probabilities of the form $P(X=k)$ , $P(X>k)$ , $P(X\leq k)$ , and $P(a\leq X\leq b)$ , and find the expected number of trials and variance in context.

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Tutorial

Modeling Real-World Trials

The geometric distribution models the number of independent trials needed to obtain the first success, where each trial has the same probability of success. To model a real-world situation with the geometric distribution, three conditions must hold:

Each trial has only two outcomes: success or failure.
The trials are independent.
The probability of success $p$ is the same on every trial.

If $X$ is the number of trials until the first success, then $X \sim \textrm{Geom}(p)$ with probability mass function

P(X = k) = (1-p)^{k-1}\, p, \quad k = 1, 2, 3, \ldots

For example, a basketball player makes free throws independently with probability $p = 0.7$ . The probability that her first made free throw is her 3rd attempt is

P(X = 3) = (0.3)^{2}(0.7) = 0.063.