5 Steps to a 5: AP Statistics 2020 by Corey Andreasen

Author:Corey Andreasen
Language: eng
Format: epub
Publisher: McGraw-Hill Education
Published: 2019-08-02T16:00:00+00:00

Geometric Distributions

In the Binomial Distributions section of this chapter, we defined a binomial setting as an experiment in which the following conditions are present:

• The experiment consists of a fixed number, n, of identical trials.

• There are only two possible outcomes: success (S) or failure (F).

• The probability of success, p, is the same for each trial.

• The trials are independent (that is, knowledge of the outcomes of earlier trials does not affect the probability of success of the next trial).

• Our interest is in a binomial random variable X, which is the count of successes in n trials. The probability distribution of X is the binomial distribution.

There are times we are interested not in the count of successes out of n fixed trials, but in the probability that the first success occurs on a given trial, or in the average number of trials until the first success. A geometric setting is defined as follows.

• There are only two possible outcomes: success (S) or failure (F).

• The probability of success, p, is the same for each trial.

• The trials are independent (that is, knowledge of the outcomes of earlier trials does not affect the probability of success of the next trial).

• Our interest is in a geometric random variable X, which is the number of trials necessary to obtain the first success.

Note that if X is a binomial, then X can take on the values 0, 1, 2, …, n. If X is geometric, then it takes on the values 1, 2, 3, …. There can be zero successes in a binomial, but the earliest a first success can come in a geometric setting is on the first trial.

If X is geometric, the probability that the first success occurs on the nth trial is given by P(X = n) = p(1 − p)n−1. The value of P(X = n) in a geometric setting can be found on the TI-83/84 calculator, in the DISTR menu, as geometpdf(p,n) (note that the order of p and n are, for reasons known only to the good folks at TI, reversed from the binomial). Given the relative simplicity of the formula for P(X = n) for a geometric setting, it’s probably just as easy to calculate the expression directly. There is also a geometcdf function that behaves analogously to the binomcdf function, but is not much needed in this course.

example: Remember Maria, the basketball player whose free-throw shooting percentage was 0.65? What is the probability that the first free throw she manages to hit is on her fourth attempt?

solution: P(X = 4) = (0.65) (1 − 0.65)4–1 = (0.65) (0.35)3 = 0.028. This can be done on the TI-83/84 as follows: geometpdf(p,n) = geometpdf(0.65,4) = 0.028.

example: In a standard deck of 52 cards, there are 12 face cards. So the probability of drawing a face card from a full deck is 12/52 = 0.231.

(a) If you draw cards with replacement (that is, you replace the card in the deck before drawing the next card), what

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