Please do it on word

3. (10 marks) Assume you have applied for two scholarships, a Merit scholarship (M) and an Athletic scholarship (A). The probability that you receive an Athletic scholarship is 0.18. The probability of receiving both scholarships is 0.11. The probability of getting at least one of the scholarships is 0.3.

a.What is the probability that you will receive a Merit scholarship?

b.b. Are events A and M mutually exclusive? Why or why not? Explain.

c.Are the two events A, and M, independent? Explain, using probabilities.

d.d. What is the probability of receiving the Athletic scholarship given that you have been awarded the Merit scholarship?

e.What is the probability of receiving the Merit scholarship given that you have been awarded the Athletic scholarship?

4. General Hospital has noted that they admit an average of 8 patients per hour.

a. What is the probability that during the next hour less than 3 patients will be admitted?

b. What is the probability that during the next two hours exactly 8 patients will be admitted?

c. What is the probability that the sample contains exactly 1 non-minority?

d. What is the expected number of minorities in the sample?

6. (12 points) Z is a standard normal variable. Find the value of Z in the following

a. The area to the left of Z is 0.8554

b. The area to the right of Z is 0.1112.

c. The area to the left of -Z is 0.0681.

d. The area to the right of -Z is 0.9803.

e. The area between 0 and Z is 0.4678.

f. The area between -Z and Z is 0.754.

7. (8 points) The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 and a standard deviation of 5,000 miles.

a. What is the probability that a randomly selected tire will have a life of no more than 50,000 miles?

b. What is the probability that a randomly selected tire will have a life of at least 47,500 miles?

c. What percentage of tires will have a life of 34,000 to 46,000 miles?

d. What is the probability that a randomly selected tire will have a life of exactly 47,500 miles?

8. (4 points) The time it takes to completely tune an engine of an automobile follows an exponential distribution with a mean of 40 minutes.

a. What is the probability of tuning an engine in 30 minutes or less?

b. What is the probability of tuning an engine between 30 and 35 minutes?