1. A class of 50 students obtained the following grades in a Statistics course:

A 5 B 25 C 15 D 5

If one student is selected at random, what is the probability that

a. The student obtained a B ?

b. The student obtained either a A or a C ?

c. The student did not obtain a D ?

d. The student did not obtain either a B or a C ?

2. A realty company has 100 homes listed for sale. Some of these homes have fireplaces, some have garages, and some have neither. A breakdown is provided in the two way classification table below.

                      Fireplace        No Fireplace
                          (F)                   (F)

Garage (G)          50                   30
No Garage (G)     10                   10

A. If one of these 100 homes is selected at random,

i. What is the probability that the home selected does not have a fireplace ?

ii. What is the probability that the home selected has a fireplace or  a garage ?

iii. What is the probability that the home selected has neither a fireplace nor a garage ?

iv. What is the probability that the home selected has a garage but not a fireplace ?

v. What is the probability that the home selected has a garage, given that it has a fireplace?

B. Are the events F = Fireplace and G = Garage independent? Show your proof.

C. Are the events F and G mutually exclusive? Why?

3. A person has applied for positions at company A, company B and company C. The probability of obtaining an offer from company A is 0.4, from company B is 0.3 and from C is 0.6. Assume that the three job offers will be independent.

a. What is the probability that the person will receive job offers  from all three companies?

b. What is the probability that the person will receive a job offer from company B only.

c. What is the probability that the person will receive a job offer from at least one of the three companies?

d. What is the probability that the person will receive exactly one job offer from the three companies?

HINT: Construct a tree diagram

4. A city council consists of 7 males and 4 females. Two councilors are selected at random to represent the city at an Elton John concert. What is the probability that

a. Both councilors are male?
b. One councilor is a male and one a female?
c. At least one councilor is a female?

NOTE: Express probabilities to 4 decimal places of accuracy. HINT: Construct a tree diagram.

5. A machine used in a production process is set up correctly 97% of the time. Given that the machine is set up correctly, only 5% of the items it produces are defective, while 95% are not defective. On the other hand, if the machine is set up incorrectly, 60% of the items it produces are defective, while only 40% are not defective. What is the probability that a randomly selected item from the production process will not be defective? NOTE: Express probabilities to 4 decimal places of accuracy.

HINT: Construct a tree diagram.

6. In a survey of 100 business executives, 60 percent were male and 40 percent were female. Seventy percent of the male executives used a PDA (personal digital assistant). Overall, 60% of the executives used a PDA. Construct a two-way classification table for the survey results.

7. You are given the probabilities listed below.

P(A) = 0.25
P(B) = 0.30
P(C) = 0.55
P( A and C ) = 0.05
P( B and C ) = 0
P(B|A) = 0.48

a. Are A and B independent events? Why or why not?
b. Find P(A and B ).
c. Find P(A or B )
d. Find P(A|B).
e. Are B and C mutually exclusive events? Why or y not?
f. Are B and C independent events? Provide a mathematical proof.