91Ó°ÊÓ

Pollution of the rivers in the United States has been a problem for many years. Consider the following events: \(A=(\) The river is polluted. \(\\}\) \(B=\\{A\) sample of water tested detects pollution. \(\\}\) \(C=\\{\) Fishing permitted. \(\\}\) Assume \(P(A)=0.3 . P(B \mid A)=0.75, P\left(B \mid A^{\prime}\right)=0.20,\) \(P(C \mid A \cap B)=0.20, P\left(C \mid A^{\prime} \cap B\right)=0.15, P\left(C \mid A \cap B^{\prime}\right)=\) \(0.80,\) and \(\mathrm{P}\left(\mathrm{C} \mid \mathrm{A}^{\prime} \mathrm{n} B^{\prime}\right)=0.90 .\) (a) Find \(P(A \cap B \cap C)\). (b) Find \(P\left(B^{\prime} \cap C\right)\). (c) Find \(P(C)\). (d) Find the probability that the river is polluted, given that fishing is permitted and the sample tested did not detect pollution.

A paint-store chain produces and sells latex and semigloss paint. Based on long-range sales, the probability that a customer will purchase latex paint is 0.75. Of those that purchase latex paint, \(60 \%\) also purchase rollers. But only \(30 \%\) of semigloss paint buyers purchase rollers. A randomly selected buyer purchases a roller and a can of paint. What is the probability that the paint is latex?

An allergist claims that \(50 \%\) of the patients she tests are allergic to some type of weed. What is the probability that (a) exactly 3 of her next 4 patients are allergic to weeds? (b) none of her next 4 patients is allergic to weeds?

By comparing appropriate regions of Venn diagrams, verify that (a) \((A \cap B) \cup\left(A n B^{\prime}\right)=A\); (b) \(\mathrm{A}^{\prime} n\left(B^{\prime} \cup C\right)=\left(A^{\prime} \mathrm{n} B^{\prime}\right) \mathrm{u}\left(A^{\prime} n C\right)\)

The probabilities that a service station will pump gas into \(0,1,2,3,4,\) or 5 or more cars during a certain 30 -minute period are 0.03,0.18,0.24,0.28 , \(0.10,\) and \(0.17,\) respectively. Find the probability that in this 30 -minute period (a) more than 2 cars receive gas; (b) at most 4 cars receive gas; (c) 4 or more cars receive gas.

How many bridge hands are possible containing 4 spades, 6 diamonds, 1 club, and 2 hearts?

If the probability is 0.1 that a person will make a mistake on his or her state income tax return, find the probability that (a) four totally unrelated persons each make a mistake; (b) Mr. Jones and Ms. Clark both make a mistake, and Mr. Roberts and Ms. Williams do not make a mistake.

A large industrial firm uses 3 local motels to provide overnight accommodations for its clients. From past experience it is known that \(20 \%:\) of the clients are assigned rooms at the Ramada Inn, \(50 \%\) at. the Sheraton, and \(30 \%\) at the Lake view Motor Lodge. If the plumbing is faulty in \(5 \%\) of the rooms at the Ramada Inn, in \(4 \%\) of the rooms at the Sheraton, and in \(8 \%\) of the rooms at the Lakeview Motor Lodge, what is the probability that (a) a client will be assigned a room with faulty plumbing? (b) a person with a room having faulty plumbing was assigned accommodations at the Lakeview Motor Lodge?

From a group of 4 men and 5 women, how many committees of size 3 are possible (a) with no restrictions? (b) with 1 man and 2 women? (c) with 2 men and 1 woman if a certain man must be on the committee?

From 4 red, 5 green, and 6 yellow apples, how many selections of 9 apples are possible if 3 of each color are to be selected?