Mathfolis

Geometric Distribution

Unit 4 · Probability, Random Variables & Distributions

What AP Stats asks here

The geometric distribution gives the number of independent Bernoulli trials needed to see the first success. The contrast with binomial is the only commonly tested decision: binomial fixes the number of trials and counts successes; geometric leaves the number of trials variable and stops at the first success.

BITS conditions

Binary
success or failure on each trial\text{success or failure on each trial}
Independent
trials do not influence each other\text{trials do not influence each other}
Trials until first success
Y∈{1,2,3,…}Y \in \{1, 2, 3, \ldots\}
Same probability
p constant on every trialp \text{ constant on every trial}

Geometric formulas

PMF
P(Y=k)=(1−p)k−1pP(Y = k) = (1 - p)^{k - 1} p
Tail (survival)
P(Y>k)=(1−p)kP(Y > k) = (1 - p)^k
Mean
μY=1p\mu_Y = \dfrac{1}{p}
SD
σY=1−pp\sigma_Y = \dfrac{\sqrt{1 - p}}{p}
AP Tip: Memory aid: binomial = 'how many successes in n tries'. Geometric = 'how many tries until the first success'. n fixed vs n variable.
Caution: When p is small, both μ and σ scale like 1/p — large mean and large variability. Long waits are normal, not a sign of bias.
Type 1

Verify BITS / Binomial vs Geometric

Trials-until-first-success ⇒ geometric. Fixed N counting successes ⇒ binomial.

Example 1
A salesperson keeps calling cold leads until making the first sale; let Y = the number of calls. Which distribution describes Y? (A) Binomial — every call has the same probability of success. (B) Geometric — counts trials until the first success. (C) Normal — the call count is symmetric. (D) Poisson — counts events in a time interval.

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Type 2

Compute P(Y=k)P(Y = k) and P(Y>k)P(Y > k)

P(Y=k)=(1−p)k−1pP(Y = k) = (1-p)^{k-1} p. The survival formula P(Y>k)=(1−p)kP(Y > k) = (1-p)^k is the cleanest way to handle 'more than k'.

Example 2
For Y∼Geom(p=0.2)Y \sim \text{Geom}(p = 0.2), what is P(Y>5)P(Y > 5)? (A) 0.20 (B) 0.33 (C) ≈ 0.3277 (D) ≈ 0.6723

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Type 3

Mean and SD

μ=1/p\mu = 1/p is the long-run average wait time. The SD grows with 1−p/p\sqrt{1-p}/p — large variability when pp is small.

Example 3
A salesperson succeeds on 5% of cold calls. On average, how many calls does it take to get the first sale? (A) 5 (B) 10 (C) 20 (D) 50

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