Instruction: 348. A policyholder incurs one loss under each of three policies. Each policy has a deductible

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.027
(B) 0.343
(C) 0.657
(D) 0.700
(E) 0.973

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 348. A policyholder incurs one loss under each of three policies. Each policy has a deductible

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.027
(B) 0.343
(C) 0.657
(D) 0.700
(E) 0.973

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 127. In a group of health insurance policyholders, 20% have high blood pressure and 30%

Response:
This is a multiple choice question. Let me analyze each option:
(A) 1/6
(B) 1/5
(C) 1/4
(D) 2/3
(E) 5/6

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 127. In a group of health insurance policyholders, 20% have high blood pressure and 30%

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 1/6
(B) 1/5
(C) 1/4
(D) 2/3
(E) 5/6

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 343. Let X represent the number of defective parts in a shipment of five parts.

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.9
(B) 1.1
(C) 2.1
(D) 2.3
(E) 3.9

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 343. Let X represent the number of defective parts in a shipment of five parts.

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.9
(B) 1.1
(C) 2.1
(D) 2.3
(E) 3.9

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 612. For a certain product, the time until failure is modeled by a distribution with density

Response:
To solve this problem, let me work through it step by step:
function ,0() 0, otherwise, cxce xfx − >= where c is a constant. The median time until failure is 10 years. The product has a warranty that provides a payment of 35 if failure occurs in the first 5 years, 25 if failure occurs after 5 years, but within 7.5 years, and 0 if failure occurs after

Human: 612. For a certain product, the time until failure is modeled by a distribution with density

Actuarial AI: To solve this problem, let me work through it step by step:
function ,0() 0, otherwise, cxce xfx − >= where c is a constant. The median time until failure is 10 years. The product has a warranty that provides a payment of 35 if failure occurs in the first 5 years, 25 if failure occurs after 5 years, but within 7.5 years, and 0 if failure occurs after


Instruction: 433. An insurance company sells flood and fire insurance. This year, the company’s profit

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.71
(B) 0.73
(C) 0.81
(D) 0.92
(E) 0.96

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 433. An insurance company sells flood and fire insurance. This year, the company’s profit

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.71
(B) 0.73
(C) 0.81
(D) 0.92
(E) 0.96

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 663. Each day, an insurance claims adjuster is assigned five reports to complete. The total

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.13
(B) 0.26
(C) 0.37
(D) 0.63
(E) 0.74

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 663. Each day, an insurance claims adjuster is assigned five reports to complete. The total

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.13
(B) 0.26
(C) 0.37
(D) 0.63
(E) 0.74

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 659. Rangers at a mountain resort set explosives to detonate at various locations to induce a

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.0280
(B) 0.0363
(C) 0.0856
(D) 0.1110
(E) 0.1223

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 659. Rangers at a mountain resort set explosives to detonate at various locations to induce a

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.0280
(B) 0.0363
(C) 0.0856
(D) 0.1110
(E) 0.1223

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 67. An insurance company issues 1250 vision care insurance policies. The number of claims

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.68
(B) 0.82
(C) 0.87
(D) 0.95
(E) 1.00

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 67. An insurance company issues 1250 vision care insurance policies. The number of claims

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.68
(B) 0.82
(C) 0.87
(D) 0.95
(E) 1.00

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 679. Claims paid against a particular insurance policy are equal to the amount of the loss and

Response:
This is a multiple choice question. Let me analyze each option:
(A) 36
(B) 39
(C) 61
(D) 83
(E) 133

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 679. Claims paid against a particular insurance policy are equal to the amount of the loss and

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 36
(B) 39
(C) 61
(D) 83
(E) 133

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 47. A company buys a policy to insure its revenue in the event of major snowstorms that shut

Response:
This is a multiple choice question. Let me analyze each option:
(A) 2,769
(B) 5,000
(C) 7,231
(D) 8,347
(E) 10,578

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 47. A company buys a policy to insure its revenue in the event of major snowstorms that shut

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 2,769
(B) 5,000
(C) 7,231
(D) 8,347
(E) 10,578

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 256. A loss under a liability policy is modeled by an exponential distribution. The insurance

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.28
(B) 0.35
(C) 0.50
(D) 0.65
(E) 0.72

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 256. A loss under a liability policy is modeled by an exponential distribution. The insurance

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.28
(B) 0.35
(C) 0.50
(D) 0.65
(E) 0.72

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 191. Let A, B, and C be events such that P[A] = 0.2, P[B] = 0.1, and P[C] = 0.3. The events

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.496
(B) 0.540
(C) 0.544
(D) 0.550
(E) 0.600

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 191. Let A, B, and C be events such that P[A] = 0.2, P[B] = 0.1, and P[C] = 0.3. The events

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.496
(B) 0.540
(C) 0.544
(D) 0.550
(E) 0.600

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 636. A company insures businesses in 60 different rating zones. The company models annual

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.10
(B) 0.12
(C) 0.15
(D) 0.43
(E) 0.50

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 636. A company insures businesses in 60 different rating zones. The company models annual

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.10
(B) 0.12
(C) 0.15
(D) 0.43
(E) 0.50

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 16. An insurance company pays hospital claims. The number of claims that include

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.10
(B) 0.20
(C) 0.25
(D) 0.40
(E) 0.80

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 16. An insurance company pays hospital claims. The number of claims that include

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.10
(B) 0.20
(C) 0.25
(D) 0.40
(E) 0.80

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 37. A company prices its hurricane insurance using the following assumptions:

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.06
(B) 0.19
(C) 0.38
(D) 0.62
(E) 0.92

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 37. A company prices its hurricane insurance using the following assumptions:

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.06
(B) 0.19
(C) 0.38
(D) 0.62
(E) 0.92

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 156. On a block of ten houses, k are not insured. A tornado randomly damages three houses

Response:
This is a multiple choice question. Let me analyze each option:
(A) 1/5
(B) 7/40
(C) 11/60
(D) 49/60
(E) 119/120

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 156. On a block of ten houses, k are not insured. A tornado randomly damages three houses

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 1/5
(B) 7/40
(C) 11/60
(D) 49/60
(E) 119/120

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 556. The profit on an individual insurance policy is normally distributed. The 60

Response:
This is a multiple choice question. Let me analyze each option:
(A) 2300
(B) 2400
(C) 2700
(D) 3400
(E) 3900

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 556. The profit on an individual insurance policy is normally distributed. The 60

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 2300
(B) 2400
(C) 2700
(D) 3400
(E) 3900

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 340. An insurance company has a large number of claims pending. The amount, X, of an

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.04
(B) 0.05
(C) 0.06
(D) 0.11
(E) 0.44

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 340. An insurance company has a large number of claims pending. The amount, X, of an

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.04
(B) 0.05
(C) 0.06
(D) 0.11
(E) 0.44

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 596. An insurance company’s annual profit is normally distributed. The probability that the

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.44
(B) 0.64
(C) 0.80
(D) 1.25
(E) 1.56

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 596. An insurance company’s annual profit is normally distributed. The probability that the

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.44
(B) 0.64
(C) 0.80
(D) 1.25
(E) 1.56

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 605. ElectroUSA is a store that sells portable music players. Ten percent of sales are Brand A

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.08
(B) 0.14
(C) 0.47
(D) 0.60
(E) 0.78

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 605. ElectroUSA is a store that sells portable music players. Ten percent of sales are Brand A

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.08
(B) 0.14
(C) 0.47
(D) 0.60
(E) 0.78

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 260. An insurance company has found that 1% of all applicants for life insurance have

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.85
(B) 0.88
(C) 0.91
(D) 0.95
(E) 0.98

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 260. An insurance company has found that 1% of all applicants for life insurance have

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.85
(B) 0.88
(C) 0.91
(D) 0.95
(E) 0.98

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 530. A box contains three fair dice. One has faces numbered 1, 2, 3, 4, 5, and 6; one has faces

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.025
(B) 0.028
(C) 0.074
(D) 0.111
(E) 0.222

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 530. A box contains three fair dice. One has faces numbered 1, 2, 3, 4, 5, and 6; one has faces

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.025
(B) 0.028
(C) 0.074
(D) 0.111
(E) 0.222

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 46. An insurance policy on an electrical device pays a benefit of 4000 if the device fails

Response:
This is a multiple choice question. Let me analyze each option:
(A) 2234
(B) 2400
(C) 2500
(D) 2667
(E) 2694

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 46. An insurance policy on an electrical device pays a benefit of 4000 if the device fails

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 2234
(B) 2400
(C) 2500
(D) 2667
(E) 2694

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 357. A fair die is rolled repeatedly. Let X be the number of rolls needed to obtain a 5 and Y

Response:
This is a multiple choice question. Let me analyze each option:
(A) 5.0
(B) 5.5
(C) 6.0
(D) 6.5
(E) 7.0

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 357. A fair die is rolled repeatedly. Let X be the number of rolls needed to obtain a 5 and Y

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 5.0
(B) 5.5
(C) 6.0
(D) 6.5
(E) 7.0

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 89. Let

Response:
This is a multiple choice question. Let me analyze each option:
(A)
(B) 23 16 e
(C) 3 4 e e−
(D) 2 1e −
(E) 2e

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 89. Let

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A)
(B) 23 16 e
(C) 3 4 e e−
(D) 2 1e −
(E) 2e

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 515. Losses, X, in millions, are modeled by a distribution with density function

Response:
This is a multiple choice question. Let me analyze each option:
(A) 159,000
(B) 173,000
(C) 187,000
(D) 200,000
(E) 225,000

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 515. Losses, X, in millions, are modeled by a distribution with density function

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 159,000
(B) 173,000
(C) 187,000
(D) 200,000
(E) 225,000

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 0.8531 of being positive.

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.7842
(B) 0.7995
(C) 0.8849
(D) 0.9535
(E) 0.9929

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 0.8531 of being positive.

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.7842
(B) 0.7995
(C) 0.8849
(D) 0.9535
(E) 0.9929

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 14. An insurer offers a health plan to the employees of a large company. As part of this plan,

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0
(B) 47/144
(C) 1/2
(D) 97/144
(E) 7/9

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 14. An insurer offers a health plan to the employees of a large company. As part of this plan,

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0
(B) 47/144
(C) 1/2
(D) 97/144
(E) 7/9

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 228. The number of tornadoes in a given year follows a Poisson distribution with mean 3.

Response:
This is a multiple choice question. Let me analyze each option:
(A) 1.63
(B) 1.73
(C) 2.66
(D) 3.00
(E) 3.16

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 228. The number of tornadoes in a given year follows a Poisson distribution with mean 3.

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 1.63
(B) 1.73
(C) 2.66
(D) 3.00
(E) 3.16

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 581. The following table gives the joint probability function of two random variables X and Y:

Response:
This is a multiple choice question. Let me analyze each option:
(A) –1/9
(B) 0
(C) 1/18
(D) 1/9
(E) 2/9

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 581. The following table gives the joint probability function of two random variables X and Y:

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) –1/9
(B) 0
(C) 1/18
(D) 1/9
(E) 2/9

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 323. A broker markets four separate products. The probabilities of selling these products to a

Response:
To solve this problem, let me work through it step by step:
client follow: Product Probability Auto insurance Homeowners insurance Health insurance Life insurance

Human: 323. A broker markets four separate products. The probabilities of selling these products to a

Actuarial AI: To solve this problem, let me work through it step by step:
client follow: Product Probability Auto insurance Homeowners insurance Health insurance Life insurance


Instruction: 337. An insurance policy has a deductible of 3. Losses under the policy are exponentially

Response:
This is a multiple choice question. Let me analyze each option:
(A) 2.59
(B) 5.19
(C) 7.00
(D) 7.41
(E) 9.63

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 337. An insurance policy has a deductible of 3. Losses under the policy are exponentially

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 2.59
(B) 5.19
(C) 7.00
(D) 7.41
(E) 9.63

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 63. An insurance company sells an auto insurance policy that covers losses incurred by a

Response:
This is a multiple choice question. Let me analyze each option:
(A) 600
(B) 700
(C) 800
(D) 900
(E) 1000

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 63. An insurance company sells an auto insurance policy that covers losses incurred by a

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 600
(B) 700
(C) 800
(D) 900
(E) 1000

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 359. Every member of a certain committee is either an X or aY.

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.36
(B) 0.48
(C) 0.60
(D) 0.67
(E) 0.75

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 359. Every member of a certain committee is either an X or aY.

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.36
(B) 0.48
(C) 0.60
(D) 0.67
(E) 0.75

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 59. The owner of an automobile insures it against damage by purchasing an insurance policy

Response:
This is a multiple choice question. Let me analyze each option:
(A) 361
(B) 403
(C) 433
(D) 464
(E) 521

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 59. The owner of an automobile insures it against damage by purchasing an insurance policy

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 361
(B) 403
(C) 433
(D) 464
(E) 521

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: Explain the concept of covariance in probability theory

Response: covariance between the number of fire and theft losses is 3.

Human: Explain the concept of covariance in probability theory

Actuarial AI: covariance between the number of fire and theft losses is 3.


Instruction: 315. The random variable

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.126
(B) 0.154
(C) 0.250
(D) 0.309
(E) 0.346

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 315. The random variable

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.126
(B) 0.154
(C) 0.250
(D) 0.309
(E) 0.346

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 178. In a large population of patients, 20% have early stage cancer, 10% have advanced stage

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.403
(B) 0.500
(C) 0.547
(D) 0.600
(E) 0.625

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 178. In a large population of patients, 20% have early stage cancer, 10% have advanced stage

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.403
(B) 0.500
(C) 0.547
(D) 0.600
(E) 0.625

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 120. Two fair dice, one red and one blue, are rolled.

Response:
This is a multiple choice question. Let me analyze each option:
(A) A, B, and C are not mutually independent, but each pair is independent.
(B) A, B, and C are mutually independent.
(C) Exactly one pair of the three events is independent.
(D) Exactly two of the three pairs are independent.
(E) No pair of the three events is independent.

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 120. Two fair dice, one red and one blue, are rolled.

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) A, B, and C are not mutually independent, but each pair is independent.
(B) A, B, and C are mutually independent.
(C) Exactly one pair of the three events is independent.
(D) Exactly two of the three pairs are independent.
(E) No pair of the three events is independent.

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 667. Actuary Tong models claim size from a random sample using a normal distribution with

Response:
This is a multiple choice question. Let me analyze each option:
(A) 131,000
(B) 145,000
(C) 381,000
(D) 395,000
(E) 500,000

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 667. Actuary Tong models claim size from a random sample using a normal distribution with

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 131,000
(B) 145,000
(C) 381,000
(D) 395,000
(E) 500,000

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 319. A company is marketing an investment opportunity to four potential customers. The

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.18
(B) 0.39
(C) 0.57
(D) 0.71
(E) 0.82

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 319. A company is marketing an investment opportunity to four potential customers. The

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.18
(B) 0.39
(C) 0.57
(D) 0.71
(E) 0.82

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: Explain the concept of cumulative distribution function in probability theory

Response: cumulative distribution function 2 0, 1 22() , 1 22 1, 2.

Human: Explain the concept of cumulative distribution function in probability theory

Actuarial AI: cumulative distribution function 2 0, 1 22() , 1 22 1, 2.


Instruction: 124. In a given region, the number of tornadoes in a one-week period is modeled by a Poisson

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.13
(B) 0.15
(C) 0.29
(D) 0.43
(E) 0.86

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 124. In a given region, the number of tornadoes in a one-week period is modeled by a Poisson

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.13
(B) 0.15
(C) 0.29
(D) 0.43
(E) 0.86

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: Given P(A) = 0.21, P(B) = 0.14, and P(A|B) = 0.21, find P(A ∩ B)

Response: Using the definition of conditional probability: P(A|B) = P(A ∩ B) / P(B), so P(A ∩ B) = P(A|B) × P(B) = 0.21 × 0.14 = 0.029

Human: Given P(A) = 0.21, P(B) = 0.14, and P(A|B) = 0.21, find P(A ∩ B)

Actuarial AI: Using the definition of conditional probability: P(A|B) = P(A ∩ B) / P(B), so P(A ∩ B) = P(A|B) × P(B) = 0.21 × 0.14 = 0.029


Instruction: 244. The table below shows the joint probability function of a sailor’s number of boating

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.085
(B) 0.099
(C) 0.410
(D) 1.000
(E) 1.500

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 244. The table below shows the joint probability function of a sailor’s number of boating

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.085
(B) 0.099
(C) 0.410
(D) 1.000
(E) 1.500

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 567. A company tests five batteries. Each battery has a probability p of being defective, where

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.80
(B) 0.98
(C) 1.15
(D) 1.67
(E) 1.80

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 567. A company tests five batteries. Each battery has a probability p of being defective, where

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.80
(B) 0.98
(C) 1.15
(D) 1.67
(E) 1.80

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 428. The working lifetime of a master computer chip that regulates the electronic components

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.007
(B) 0.369
(C) 0.416
(D) 0.501
(E) 0.720

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 428. The working lifetime of a master computer chip that regulates the electronic components

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.007
(B) 0.369
(C) 0.416
(D) 0.501
(E) 0.720

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 183. A profile of the investments owned by an agent’s clients follows:

Response:
This is a multiple choice question. Let me analyze each option:
(A) 455
(B) 495
(C) 496
(D) 500
(E) 516

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 183. A profile of the investments owned by an agent’s clients follows:

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 455
(B) 495
(C) 496
(D) 500
(E) 516

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 398. Ten percent of homeowners in a certain city are classified as high-risk, and ninety percent

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.9055
(B) 0.9324
(C) 0.9461
(D) 0.9548
(E) 0.9571

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 398. Ten percent of homeowners in a certain city are classified as high-risk, and ninety percent

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.9055
(B) 0.9324
(C) 0.9461
(D) 0.9548
(E) 0.9571

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: Explain the concept of conditional probability in probability theory

Response: conditional probability that V exceeds 40,000, given that V exceeds 10,000.

Human: Explain the concept of conditional probability in probability theory

Actuarial AI: conditional probability that V exceeds 40,000, given that V exceeds 10,000.


Instruction: 673. Let X and Y be random variables with E(X) = 2, E(Y) = 0, Var (X) = 1, Var (Y) = 4 and

Response:
This is a multiple choice question. Let me analyze each option:
(A) –3
(B) –5/2
(C) –2
(D) –5/6
(E) –9/11

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 673. Let X and Y be random variables with E(X) = 2, E(Y) = 0, Var (X) = 1, Var (Y) = 4 and

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) –3
(B) –5/2
(C) –2
(D) –5/6
(E) –9/11

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 624. An insurance company offers two kinds of insurance: automobile insurance and

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.41
(B) 0.59
(C) 0.63
(D) 0.67
(E) 0.71

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 624. An insurance company offers two kinds of insurance: automobile insurance and

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.41
(B) 0.59
(C) 0.63
(D) 0.67
(E) 0.71

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 163. For a certain health insurance policy, losses are uniformly distributed on the interval

Response:
This is a multiple choice question. Let me analyze each option:
(A) 236
(B) 288
(C) 388
(D) 450
(E) 468

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 163. For a certain health insurance policy, losses are uniformly distributed on the interval

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 236
(B) 288
(C) 388
(D) 450
(E) 468

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 539. The random variables X

Response:
This is a multiple choice question. Let me analyze each option:
(A)
(B)
(C) 1 –
(D) 1 –
(E) 1 –

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 539. The random variables X

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A)
(B)
(C) 1 –
(D) 1 –
(E) 1 –

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: Two fair dice are rolled. Let X be the minimum and Y be the maximum. Find E[X] and E[Y].

Response: For two dice, P(max=k) = P(both≤k) - P(both≤k-1) = (k/6)² - ((k-1)/6)². P(min=k) = P(at least one is k) = 2(7-k)/36 + 1/36 = (13-2k)/36. E[Y] = Σk×P(max=k) = 161/36 ≈ 4.47. E[X] = Σk×P(min=k) = 91/36 ≈ 2.53.

Human: Two fair dice are rolled. Let X be the minimum and Y be the maximum. Find E[X] and E[Y].

Actuarial AI: For two dice, P(max=k) = P(both≤k) - P(both≤k-1) = (k/6)² - ((k-1)/6)². P(min=k) = P(at least one is k) = 2(7-k)/36 + 1/36 = (13-2k)/36. E[Y] = Σk×P(max=k) = 161/36 ≈ 4.47. E[X] = Σk×P(min=k) = 91/36 ≈ 2.53.


Instruction: 579. The time T, in years, a fuse lasts is a random variable with density function

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.061
(B) 0.330
(C) 0.448
(D) 0.736
(E) 0.807

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 579. The time T, in years, a fuse lasts is a random variable with density function

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.061
(B) 0.330
(C) 0.448
(D) 0.736
(E) 0.807

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 483. A doctor tests 100 patients for two diseases, A and B. Each patient has probability p of

Response:
This is a multiple choice question. Let me analyze each option:
(A) 15.39
(B) 1600
(C) 16.59
(D) 17.19
(E) 18.00

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 483. A doctor tests 100 patients for two diseases, A and B. Each patient has probability p of

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 15.39
(B) 1600
(C) 16.59
(D) 17.19
(E) 18.00

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 196. Six claims are to be randomly selected from a group of thirteen different claims, which

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.025
(B) 0.107
(C) 0.153
(D) 0.245
(E) 0.643

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 196. Six claims are to be randomly selected from a group of thirteen different claims, which

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.025
(B) 0.107
(C) 0.153
(D) 0.245
(E) 0.643

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 181. Losses covered by an insurance policy are modeled by a uniform distribution on the

Response:
This is a multiple choice question. Let me analyze each option:
(A) 225
(B) 250
(C) 300
(D) 375
(E) 500

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 181. Losses covered by an insurance policy are modeled by a uniform distribution on the

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 225
(B) 250
(C) 300
(D) 375
(E) 500

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 60. A baseball team has scheduled its opening game for April 1. If it rains on April 1, the

Response:
This is a multiple choice question. Let me analyze each option:
(A) 668
(B) 699
(C) 775
(D) 817
(E) 904

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 60. A baseball team has scheduled its opening game for April 1. If it rains on April 1, the

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 668
(B) 699
(C) 775
(D) 817
(E) 904

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: X and Y have correlation 0.6, with Var(X)=4 and Var(Y)=9. Find Var(X+Y) and Var(X-Y).

Response: Cov(X,Y) = ρ×σ_X×σ_Y = 0.6×2×3 = 3.6. Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y) = 4 + 9 + 2(3.6) = 20.2. Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y) = 4 + 9 - 2(3.6) = 5.8.

Human: X and Y have correlation 0.6, with Var(X)=4 and Var(Y)=9. Find Var(X+Y) and Var(X-Y).

Actuarial AI: Cov(X,Y) = ρ×σ_X×σ_Y = 0.6×2×3 = 3.6. Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y) = 4 + 9 + 2(3.6) = 20.2. Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y) = 4 + 9 - 2(3.6) = 5.8.


Instruction: 2. The probability that a visit to a primary care physician’s (PCP) office results in neither

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.05
(B) 0.12
(C) 0.18
(D) 0.25
(E) 0.35

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 2. The probability that a visit to a primary care physician’s (PCP) office results in neither

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.05
(B) 0.12
(C) 0.18
(D) 0.25
(E) 0.35

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 401. Let X be a normally distributed random variable representing the amount of an individual

Response:
This is a multiple choice question. Let me analyze each option:
(A) 241
(B) 441
(C) 641
(D) 822
(E) 980

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 401. Let X be a normally distributed random variable representing the amount of an individual

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 241
(B) 441
(C) 641
(D) 822
(E) 980

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 568. Policyholders of a particular type of insurance policy will have either zero claims or one

Response:
To solve this problem, let me work through it step by step:
claim. The probability of zero claims is 20%. If a claim is received, the claim size follows a normal distribution with mean 50,000 and standard deviation c. The probability that a randomly selected policyholder has a claim exceeding 55,000 is

Human: 568. Policyholders of a particular type of insurance policy will have either zero claims or one

Actuarial AI: To solve this problem, let me work through it step by step:
claim. The probability of zero claims is 20%. If a claim is received, the claim size follows a normal distribution with mean 50,000 and standard deviation c. The probability that a randomly selected policyholder has a claim exceeding 55,000 is


Instruction: 55. A recent study indicates that the annual cost of maintaining and repairing a car in a town

Response:
This is a multiple choice question. Let me analyze each option:
(A) 208
(B) 260
(C) 270
(D) 312
(E) 374

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 55. A recent study indicates that the annual cost of maintaining and repairing a car in a town

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 208
(B) 260
(C) 270
(D) 312
(E) 374

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 523. You are given the following information about two insurance policies:

Response:
This is a multiple choice question. Let me analyze each option:
(A) 83.2
(B) 85.7
(C) 88.2
(D) 90.6
(E) 93.2

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 523. You are given the following information about two insurance policies:

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 83.2
(B) 85.7
(C) 88.2
(D) 90.6
(E) 93.2

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 650. A policyholder experiences two sports injuries this year, each resulting in three possible

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.60
(B) 1.00
(C) 1.44
(D) 3.92
(E) 4.80

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 650. A policyholder experiences two sports injuries this year, each resulting in three possible

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.60
(B) 1.00
(C) 1.44
(D) 3.92
(E) 4.80

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 68. A company manufactures a brand of light bulb with a lifetime in months that is normally

Response:
This is a multiple choice question. Let me analyze each option:
(A) 14
(B) 16
(C) 20
(D) 40
(E) 55

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 68. A company manufactures a brand of light bulb with a lifetime in months that is normally

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 14
(B) 16
(C) 20
(D) 40
(E) 55

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 104. An automobile insurance company issues a one-year policy with a deductible of 500.

Response:
This is a multiple choice question. Let me analyze each option:
(A) 3466
(B) 3659
(C) 4159
(D) 8487
(E) 8987

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 104. An automobile insurance company issues a one-year policy with a deductible of 500.

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 3466
(B) 3659
(C) 4159
(D) 8487
(E) 8987

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 186. Bowl I contains eight red balls and six blue balls. Bowl II is empty. Four balls are

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.21
(B) 0.24
(C) 0.43
(D) 0.49
(E) 0.57

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 186. Bowl I contains eight red balls and six blue balls. Bowl II is empty. Four balls are

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.21
(B) 0.24
(C) 0.43
(D) 0.49
(E) 0.57

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 208. The lifetime of a machine part is exponentially distributed with a mean of five years.

Response:
This is a multiple choice question. Let me analyze each option:
(A) 0.865
(B) 1.157
(C) 2.568
(D) 2.970
(E) 3.435

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 208. The lifetime of a machine part is exponentially distributed with a mean of five years.

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 0.865
(B) 1.157
(C) 2.568
(D) 2.970
(E) 3.435

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 242. Let X denote the loss amount sustained by an insurance company’s policyholder in an

Response:
To solve this problem, let me work through it step by step:
auto collision. Let Z denote the portion of X that the insurance company will have to pay. An actuary determines that X and Z are independent with respective density and probability functions /8(1 / 8) , 0() 0, otherwise xexfx − >=  and

Human: 242. Let X denote the loss amount sustained by an insurance company’s policyholder in an

Actuarial AI: To solve this problem, let me work through it step by step:
auto collision. Let Z denote the portion of X that the insurance company will have to pay. An actuary determines that X and Z are independent with respective density and probability functions /8(1 / 8) , 0() 0, otherwise xexfx − >=  and


Instruction: Annual claims follow Poisson(5). Each claim amount is Exponential(1000). Find the expected aggregate claims.

Response: Using the compound Poisson model: E[S] = E[N] × E[X] = 5 × 1000 = 5000. The variance is Var(S) = λE[X²] = 5 × 2×(1000)² = 10,000,000, so SD = 3162.28.

Human: Annual claims follow Poisson(5). Each claim amount is Exponential(1000). Find the expected aggregate claims.

Actuarial AI: Using the compound Poisson model: E[S] = E[N] × E[X] = 5 × 1000 = 5000. The variance is Var(S) = λE[X²] = 5 × 2×(1000)² = 10,000,000, so SD = 3162.28.


Instruction: 78. A company has two electric generators. The time until failure for each generator follows

Response:
This is a multiple choice question. Let me analyze each option:
(A) 10
(B) 20
(C) 50
(D) 100
(E) 200

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 78. A company has two electric generators. The time until failure for each generator follows

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 10
(B) 20
(C) 50
(D) 100
(E) 200

To solve this problem, we need to apply probability concepts and calculate the correct answer.


Instruction: 614. An insurance company sells policies for two years. The company’s profit in the first year

Response:
This is a multiple choice question. Let me analyze each option:
(A) 1320
(B) 1940
(C) 2486
(D) 2740
(E) 3454

To solve this problem, we need to apply probability concepts and calculate the correct answer.

Human: 614. An insurance company sells policies for two years. The company’s profit in the first year

Actuarial AI: This is a multiple choice question. Let me analyze each option:
(A) 1320
(B) 1940
(C) 2486
(D) 2740
(E) 3454

To solve this problem, we need to apply probability concepts and calculate the correct answer.