1. Section 8.2: The Geometric Distribution
AP Statistics
Section 8.2: The Geometric Distribution

2. Objective: To be able to understand and calculate geometric probabilities.
Ex. Russian Roulette
Criteria for a Geometric Random Variable:
Each observation can be classified as a success or failure.
p is the probability of success and p is fixed.
The observations are independent.
The random variable measures the number of trials necessary to obtain the first success. (it includes the 1st success)

3. Geometrical Probability Formula:
If data are produced in a geometric setting and X = the number of trials until the first success occurs, then X is called a geometric random variable.
Geometrical Probability Formula:
Let X be a geometric random variable and n be the number of trials until we obtain our first success. If X~G(p), then
𝑃 𝑋=𝑛 = 𝑞 (𝑛−1) ∙𝑝
The geometric probability distribution is as follows:
𝑿= 𝒙 𝒊 1 2 3 … n
P(𝑿= 𝒙 𝒊 ) 𝒑 𝒒𝒑
𝒒 𝟐 𝒑
𝒒 𝒏−𝟏 ∙𝒑
F(𝑿= 𝒙 𝒊 )
𝒑+𝒒𝒑
𝒑+𝒒𝒑+ 𝒒 𝟐 𝒑
𝑝 𝑛

4. Points:
This is an infinite distribution.
The smallest X can be is 1.
Every geometric distribution is __________________.
The cumulative distribution is always ___________________.
Calculator notation:
For P(X = k) --- use geometpdf(p,X)
For P(X ≤ k) --- use geometcdf(p,X)
How can we show that the sum of the terms of an infinite distribution sum to 1?

5. Ex. Suppose you work at a blood bank and are interested in collecting type A blood. It is known that 15% of the population is type A. Let X represent the number of donors until and including the first type A donor is found.
Does this example meet the criteria for a geometric setting?
Find probability that the first type A donor is the 4th donor of the day.
Find probability that the first type A donor is the 2nd donor of the day.
Find probability that the first type A donor occurs before the 4th donor of the day.

6. Find probability that the first type A donor is at least the 5th donor of the day.
Complete the probability distribution for the geometric random variable X for the first 6 terms.
𝑿= 𝒙 𝒊
P(𝑿= 𝒙 𝒊 )
F(𝑿= 𝒙 𝒊 )

7. If X ~ G(p), then 𝜇 𝑋 = 1 𝑝 and 𝜎 2 𝑋 = (1−𝑝) 𝑝 2
If X ~ G(p), then 𝜇 𝑋 = 1 𝑝 and 𝜎 2 𝑋 = (1−𝑝) 𝑝 2 . Two Methods for Solving P(X > n): Proof of method #2: