178. In a large population of patients, 20% have early stage cancer, 10% have advanced stage cancer, and the other 70% do not have cancer. Six patients from this population are randomly selected.

Calculate the expected number of selected patients with advanced stage cancer, given that at least one of the selected patients has early stage cancer.
(A) 0.403
(B) 0.500
(C) 0.547
(D) 0.600
(E) 0.625

151. The following information is given about a group of high-risk borrowers.i) Of all these borrowers, 30% defaulted on at least one student loan.ii) Of the borrowers who defaulted on at least one car loan, 40% defaulted on at least one student loan.iii) Of the borrowers who did not default on any student loans, 28% defaulted on at least one car loan.A statistician randomly selects a borrower from this group and observes that the selected borrower defaulted on at least one student loan.Calculate the probability that the selected borrower defaulted on at least one car loan.(A) 0.33(B) 0.40(C) 0.44(D) 0.65(E) 0.72

152. An insurance company issues policies covering damage to automobiles. The amount of damage is modeled by a uniform distribution on [0, b]. The policy payout is subject to a deductible of b/10. A policyholder experiences automobile damage. Calculate the ratio of the standard deviation of the policy payout to the standard deviation of the amount of the damage. (A) 0.8100 (B) 0.9000 (C) 0.9477 (D) 0.9487 (E) 0.9735

153. 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: i) The policyholder has exactly one accident in year one and has more than one accident in year two. ii) The policyholder has at least two accidents during the two-year period. iii) The policyholder has exactly one accident in year one and has at least one accident in year two. iv) The policyholder has exactly one accident in year one and has a total of two or more accidents in the two-year period. v) 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 F ∩ G. (A) None (B) Exactly one (C) Exactly two (D) Exactly three (E) All

154. An insurance company categorizes its policyholders into three mutually exclusive groups: high-risk, medium-risk, and low-risk. An internal study of the company showed that 45% of the policyholders are low-risk and 35% are medium-risk. The probability of death over the next year, given that a policyholder is high-risk is two times the probability of death of a medium-risk policyholder. The probability of death over the next year, given that a policyholder is medium-risk is three times the probability of death of a low-risk policyholder. The probability of death of a randomly selected policyholder, over the next year, is 0.009. Calculate the probability of death of a policyholder over the next year, given that the policyholder is high-risk. (A) 0.0025 (B) 0.0200 (C) 0.1215 (D) 0.2000 (E) 0.3750

155. A policy covers a gas furnace for one year. During that year, only one of three problems can occur: i) The igniter switch may need to be replaced at a cost of 60. There is a 0.10 probability of this. ii) The pilot light may need to be replaced at a cost of 200. There is a 0.05 probability of this. iii) The furnace may need to be replaced at a cost of 3000. There is a 0.01 probability of this. Calculate the deductible that would produce an expected claim payment of 30. (A) 100 (B) At least 100 but less than 150 (C) At least 150 but less than 200 (D) At least 200 but less than 250 (E) At least 250

156. On a block of ten houses, k are not insured. A tornado randomly damages three houses on the block. The probability that none of the damaged houses are insured is 1/120. Calculate the probability that at most one of the damaged houses is insured. (A) 1/5 (B) 7/40 (C) 11/60 (D) 49/60 (E) 119/120

157. In a casino game, a gambler selects four different numbers from the first twelve positive integers. The casino then randomly draws nine numbers without replacement from the first twelve positive integers. The gambler wins the jackpot if the casino draws all four of the gambler’s selected numbers. Calculate the probability that the gambler wins the jackpot. (A) 0.002 (B) 0.255 (C) 0.296 (D) 0.573 (E) 0.625

158. The number of days an employee is sick each month is modeled by a Poisson distribution with mean 1. The numbers of sick days in different months are mutually independent. Calculate the probability that an employee is sick more than two days in a three-month period. (A) 0.199 (B) 0.224 (C) 0.423 (D) 0.577 (E) 0.801

159. The number of traffic accidents per week at intersection Q has a Poisson distribution with mean 3. The number of traffic accidents per week at intersection R has a Poisson distribution with mean 1.5. Let A be the probability that the number of accidents at intersection Q exceeds its mean. Let B be the corresponding probability for intersection R. Calculate B – A. (A) 0.00 (B) 0.09 (C) 0.13 (D) 0.19 (E) 0.31

160. Losses due to accidents at an amusement park are exponentially distributed. An insurance company offers the park owner two different policies, with different premiums, to insure against losses due to accidents at the park. Policy A has a deductible of 1.44. For a random loss, the probability is 0.640 that under this policy, the insurer will pay some money to the park owner. Policy B has a deductible of d. For a random loss, the probability is 0.512 that under this policy, the insurer will pay some money to the park owner. Calculate d. (A) 0.960 (B) 1.152 (C) 1.728 (D) 1.800 (E) 2.160

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