AP Statistics

• Section 8.2: The Geometric Distribution

• Objective: To be able to understand and calculate geometric probabilities.

Ex. Russian Roulette

• Criteria for a Geoemtric Random Variable:1. Each observation can be classified as a success or failure.2. p is the probability of success and p is fixed.3. The observations are independent.4. The random variable measures the number of trials

necessary to obtain the first success. (it includes the 1st success)

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

The geometric probability distribution is as follows:1 2 3 … n …

P() … …F() … …

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?

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. a. Does this example meet the criteria for a geometric setting?

b. Find probability that the first type A donor is the 4th donor of the day.

c. Find probability that the first type A donor is the 2nd donor of the day.

d. Find probability that the first type A donor occurs before the 4th donor of the day.

e. Find probability that the first type A donor is at least the 5th donor of the day.

f. Complete the probability distribution for the geometric random variable X for the first 6 terms.

P()

F()

If X ~ G(p), then and .

Two Methods for Solving P(X > n):1. 2.

Proof of method #2: