A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability 5/8, French with probability 5/8, and art and French together with probability 1/4. What is the probability that he chooses French and mathematics?

• 1/8
• 1/4
• 3/8
• 1/2
• 5/8

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

For a bill to come before the president of the United States, it must be passed by both the House of Representatives and the Senate. Assume that, of the bills presented to these two bodies, 60 percent pass the House, 80 percent pass the Senate, and 90 percent pass at least one of the two. Calculate the probability that the next bill presented to the two groups will come before the president.

• 0.25
• 0.35
• 0.4
• 0.5
• 0.6

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

A University is hosting its annual actuarial alumni event that will host 150 people. You are given the following about the attendees:

• 75 have the ASA designation
• 50 have the FSA designation
• 35 have the ACAS designation
• 30 have the FCAS designation
• 90 have two designations
• None of the attendees have more than two designations

The ASA designation is required to get the FSA designation and the ACAS designation is required to get the FCAS designation. Determine the number of attendees who are associates (ASA or ACAS) but not fellows (FSA or FCAS).

• 15
• 20
• 25
• 30
• 40

A survey of a group’s viewing habits over the last year revealed the following information:

1. 28% watched gymnastics
2. 29% watched baseball
3. 19% watched soccer
4. 14% watched gymnastics and baseball
5. 12% watched baseball and soccer
6. 10% watched gymnastics and soccer
7. 8% watched all three sports.

Calculate the percentage of the group that watched none of the three sports during the last year.

• 24%
• 36%
• 41%
• 52%
• 60%

An auto insurance company has 10,000 policyholders. Each policyholder is classified as

1. young or old;
2. male or female; and
3. married or single.

Of these policyholders, 3000 are young, 4600 are male, and 7000 are married. The policyholders can also be classified as 1320 young males, 3010 married males, and 1400 young married persons. Finally, 600 of the policyholders are young married males.

Calculate the number of the company’s policyholders who are young, female, and single.

• 280
• 423
• 486
• 880
• 896

An insurance company estimates that 40% of policyholders who have only an auto policy will renew next year and 60% of policyholders who have only a homeowners policy will renew next year. The company estimates that 80% of policyholders who have both an auto policy and a homeowners policy will renew at least one of those policies next year.

Company records show that 65% of policyholders have an auto policy, 50% of policyholders have a homeowners policy, and 15% of policyholders have both an auto policy and a homeowners policy.

Using the company’s estimates, calculate the percentage of policyholders that will renew at least one policy next year.

• 20%
• 29%
• 41%
• 53%
• 70%

Among a large group of patients recovering from shoulder injuries, it is found that 22% visit both a physical therapist and a chiropractor, whereas 12% visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.14 the probability that a patient visits a physical therapist.

Calculate the probability that a randomly chosen member of this group visits a physical therapist.

• 0.26
• 0.38
• 0.40
• 0.48
• 0.62

An insurance company examines its pool of auto insurance customers and gathers the following information:

1. All customers insure at least one car.
2. 70% of the customers insure more than one car.
3. 20% of the customers insure a sports car.
4. Of those customers who insure more than one car, 15% insure a sports car.

Calculate the probability that a randomly selected customer insures exactly one car and that car is not a sports car.

• 0.13
• 0.21
• 0.24
• 0.25
• 0.30

A survey of 100 TV viewers revealed that over the last year:

1. 34 watched CBS.
2. 15 watched NBC.
3. 10 watched ABC.
4. 7 watched CBS and NBC.
5. 6 watched CBS and ABC.
6. 5 watched NBC and ABC.
7. 4 watched CBS, NBC, and ABC.
8. 18 watched HGTV, and of these, none watched CBS, NBC, or ABC.

Calculate how many of the 100 TV viewers did not watch any of the four channels (CBS, NBC, ABC or HGTV).

• 1
• 37
• 45
• 55
• 82

The annual numbers of thefts a homeowners insurance policyholder experiences are analyzed over three years.

Define the following events:

1. A = the event that the policyholder experiences no thefts in the three years.
2. B = the event that the policyholder experiences at least one theft in the second year.
3. C = the event that the policyholder experiences exactly one theft in the first year.
4. D = the event that the policyholder experiences no thefts in the third year.
5. E = the event that the policyholder experiences no thefts in the second year, and at least one theft in the third year.

Determine which three events satisfy the condition that the probability of their union equals the sum of their probabilities.

• Events A, B, and E
• Events A, C, and E
• Events A, D, and E
• Events B, C, and D
• Events B, C, and E