A policyholder purchases automobile insurance for two years. Define the following events:

F = the policyholder has exactly one accident in year one.

G = the policyholder has one or more accidents in year two. Define the following events:

1. The policyholder has exactly one accident in year one and has more than one accident in year two.
2. The policyholder has at least two accidents during the two-year period.
3. The policyholder has exactly one accident in year one and has at least one accident in year two.
4. The policyholder has exactly one accident in year one and has a total of two or more accidents in the two-year period.
5. The policyholder has exactly one accident in year one and has more accidents in year two than in year one.

Determine the number of events from the above list of five that are the same as [math]F \cap G [/math].

• None
• Exactly one
• Exactly two
• Exactly three
• All

An actuary compiles the following information from a portfolio of 1000 homeowners insurance policies:

1. 130 policies insure three-bedroom homes.
2. 280 policies insure one-story homes.
3. 150 policies insure two-bath homes.
4. 30 policies insure three-bedroom, two-bath homes.
5. 50 policies insure one-story, two-bath homes.
6. 40 policies insure three-bedroom, one-story homes.
7. 10 policies insure three-bedroom, one-story, two-bath homes.

Calculate the number of homeowners policies in the portfolio that insure neither one-story nor two-bath nor three-bedroom homes.

• 310
• 450
• 530
• 550
• 570

In order to boost retention and recruitment, a restaurant owner has decided to offer its employees health, dental and vision insurance to its employees. Each employee is eligible for health insurance but can only choose one of vision or dental insurance. The owner has compiled the following enrolment information:

• 15% of employees opted out of any insurance coverage
• 70% of employees enrolled for health insurance coverage
• 35% of employees enrolled for health but not vision
• 10% of employees enrolled for dental but not health

Determine the percentage of employees who enrolled in vision but not dental.

1. 25%
2. 30%
3. 35%
4. 38%
5. 40%

You are given

• [math]\operatorname{P}[A \cup B] = 0.7[/math]
• [math]\operatorname{P}[A \cup B'] = 0.9[/math]

Calculate [math]\operatorname{P}[A] [/math].

• 0.2
• 0.3
• 0.4
• 0.6
• 0.8

An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is 0.44.

Calculate the number of blue balls in the second urn.

• 4
• 20
• 24
• 44
• 64

An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company’s employees that choose coverages A, B, and C are 1/4, 1/3, and 5/12 respectively. Calculate the probability that a randomly chosen employee will choose no supplementary coverage

• 0
• 47/144
• 1/2
• 97/144
• 7/9

An insurance agent offers his clients auto insurance, homeowners insurance and renters insurance. The purchase of homeowners insurance and the purchase of renters insurance are mutually exclusive. The profile of the agent’s clients is as follows:

1. 17% of the clients have none of these three products.
2. 64% of the clients have auto insurance.
3. Twice as many of the clients have homeowners insurance as have renters insurance.
4. 35% of the clients have two of these three products.
5. 11% of the clients have homeowners insurance, but not auto insurance.

Calculate the percentage of the agent’s clients that have both auto and renters insurance.

• 7%
• 10%
• 16%
• 25%
• 28%

A profile of the investments owned by an agent’s clients follows:

1. 228 own annuities.
2. 220 own mutual funds.
3. 98 own life insurance and mutual funds.
4. 93 own annuities and mutual funds.
5. 16 own annuities, mutual funds, and life insurance.
6. 45 more clients own only life insurance than own only annuities.
7. 290 own only one type of investment (i.e., annuity, mutual fund, or life insurance).

Calculate the agent’s total number of clients.

• 455
• 495
• 496
• 500
• 516

In a fierce battle, not less than 70 percent of the soldiers lost one eye and not less than 75 percent lost one ear. What is the minimal possible percentage of those who simultaneously lost one ear and one eye?

• 0.4
• 0.45
• 0.55
• 0.6
• 0.65

References

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

In a certain group of cancer patients, each patient's cancer is classified in exactly one of the following five stages: stage 0, stage 1, stage 2, stage 3, or stage 4.

1. 75% of the patients in the group have stage 2 or lower.
2. 80% of the patients in the group have stage 1 or higher.
3. 80% of the patients in the group have stage 0, 1, 3, or 4.

One patient from the group is randomly selected.

Calculate the probability that the selected patient's cancer is stage 1.

• 0.20
• 0.25
• 0.35
• 0.48
• 0.65