108. In a certain game of chance, a square board with area 1 is colored with sectors of either red or blue. A player, who cannot see the board, must specify a point on the board by giving an x-coordinate and a y-coordinate. The player wins the game if the specified point is in a blue sector. The game can be arranged with any number of red sectors, and the red sectors are designed so that 
 , where Ri is the area of the red sector.
Calculate the minimum number of red sectors that makes the chance of a player winning less than 20%.
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

101. The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of d, the insurance company reduces the expected claim payment by 10%. Calculate the percentage reduction on the variance of the claim payment. (A) 1% (B) 5% (C) 10% (D) 20% (E) 25%

102. The number of hurricanes that will hit a certain house in the next ten years is Poisson distributed with mean 4. Each hurricane results in a loss that is exponentially distributed with mean 1000. Losses are mutually independent and independent of the number of hurricanes. Calculate the variance of the total loss due to hurricanes hitting this house in the next ten years. (A) 4,000,000 (B) 4,004,000 (C) 8,000,000 (D) 16,000,000 (E) 20,000,000

103. A motorist makes three driving errors, each independently resulting in an accident with probability 0.25. Each accident results in a loss that is exponentially distributed with mean 0.80. Losses are mutually independent and independent of the number of accidents. The motorist’s insurer reimburses 70% of each loss due to an accident. Calculate the variance of the total unreimbursed loss the motorist experiences due to accidents resulting from these driving errors. (A) 0.0432 (B) 0.0756 (C) 0.1782 (D) 0.2520 (E) 0.4116

104. An automobile insurance company issues a one-year policy with a deductible of 500. The probability is 0.8 that the insured automobile has no accident and 0.0 that the automobile has more than one accident. If there is an accident, the loss before application of the deductible is exponentially distributed with mean 3000. Calculate the percentile of the insurance company payout on this policy. (A) 3466 (B) 3659 (C) 4159 (D) 8487 (E) 8987

105. From 27 pieces of luggage, an airline luggage handler damages a random sample of four. The probability that exactly one of the damaged pieces of luggage is insured is twice the probability that none of the damaged pieces are insured. Calculate the probability that exactly two of the four damaged pieces are insured. (A) 0.06 (B) 0.13 (C) 0.27 (D) 0.30 (E) 0.31

106. Automobile policies are separated into two groups: low-risk and high-risk. Actuary Rahul examines low-risk policies, continuing until a policy with a claim is found and then stopping. Actuary Toby follows the same procedure with high-risk policies. Each low-risk policy has a 10% probability of having a claim. Each high-risk policy has a 20% probability of having a claim. The claim statuses of polices are mutually independent. Calculate the probability that Actuary Rahul examines fewer policies than Actuary Toby. (A) 0.2857 (B) 0.3214 (C) 0.3333 (D) 0.3571 (E) 0.4000

107. Let X represent the number of customers arriving during the morning hours and let Y represent the number of customers arriving during the afternoon hours at a diner. You are given:i) X and Y are Poisson distributed.ii) The first moment of X is less than the first moment of Y by 8.iii) The second moment of X is 60% of the second moment of Y.Calculate the variance of Y.(A) 4(B) 12(C) 16(D) 27(E) 35

109. Automobile claim amounts are modeled by a uniform distribution on the interval [0,10,000]. Actuary A reports X, the claim amount divided by 1000. Actuary B reports Y, which is X rounded to the nearest integer from 0 to 10. Calculate the absolute value of the difference between the 4th moment of X and the 4th moment of Y. (A) 0 (B) 33 (C) 296 (D) 303 (E) 533

110. The probability of x losses occurring in year 1 is 1 for x=0,1,2, ... The probability of y losses in year 2 given x losses in year 1 is given by the table:  Calculate the probability of exactly 2 losses in 2 years. (A) 0.025 (B) 0.031 (C) 0.075 (D) 0.100 (E) 0.131

111. Let X be a continuous random variable with density function Calculate the value of p such that E(X) = 2. (A) 1 (B) 2.5 (C) 3 (D) 5 (E) There is no such p.

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