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FINA2220A Quantitative Methods for Actuarial Analysis

First Term 2023-2024

Assignment 3

Hand in the solutions on or before 19 October 2023.

1.    Two balls are chosen randomly from an urn containing 5 white, 4 black, and 3 orange balls. Suppose that we win $3 for each black ball selected and we lose $2 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value?

2.    A gambling book recommends the following “winning strategy” of the game of roulette.  It recommends that a gambler bet $1 on red. If red appears (which has probability 18/38), then the gambler should take her $1 profit and quit. If the gambler loses this bet (which has probability 20/38 of occurring), she should make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote the gambler’s winnings when she quits.

(a)  Find P(X > 0).

(b)  Are you convinced that the strategy is indeed a “winning” strategy? Explain your answer! (c)  Find E(X).

3.    A box contains 7 red and 3 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $1.20; if they are different colors, then you win − $0.90 (that is, you lose $0.90). Calculate

(a)  the expected value of the amount you win;

(b)  the variance of the amount you win.

4.    Two players put one dollar into a pot. They decide to throw a pair of dice alternately. The first one who throws a total of 6 on both dice wins the pot. How much should the player who starts add to the pot to make this a fair game?

5.    If E(X) = 2 and var(X) = 7, find

(a)   E {(3 + X)3} .

(b)   var (5 + 7X) .

6.    On a multiple-choice exam with 4 possible answers for each of the  8 questions, what is the probability that a student would get 6 or more correct answers just by guessing?

7.    A hospital receives 1/4 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company X’s shipments, 8% of the vials are ineffective. For every other company, 2% of its vials are ineffective. The hospital tests 20 randomly selected vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X?

8.    Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ = 3.

(a)  Find the probability that 4 or more accidents occur today.

(b)  Repeat part (a) under the assumption that at least 1 accident occurs today.

9.    The probability of being dealt a full house in a hand of poker is approximately 0.0014. Find an approximation for the probability that in 3000 hands of poker you will be dealt at least 3 full houses.

10.  An actuarial student, Margret, injures herself at a rate of 2 times per month. Suppose that the

Poisson distribution models the number of times that she injures herself.

(a)  Calculate the probability that she injures herself exactly 1 time next month. (b)  Calculate the probability that she injures herself exactly 8 times next year.

11.  A company buys a policy to insure its revenue in the event of major snowstorms that shut down business. The policy pays nothing for the first such snowstorm of the year and $20000 for each snowstorm thereafter, until the end of the year. The number of major snowstorms per year that shut down business is assumed to have a Poisson distribution with mean 2.5. What is the expected amount paid to the company under this policy during a one-year period?

12.  An insurance policy on an electrical device pays a benefit of 2000 if the device fails during the first year. The amount of the benefit decreases by 500 each successive year until it reaches 0. If the device has not failed by the beginning of any given year, the probability of failure during that year is 0.3. What is the expected benefit under this policy?

13.  Two athletic teams play a series of games; the first team to win 3 games is declared the overall winner. Suppose that one of the teams is stronger than the other and wins each game with probability 0.55, independently of the outcomes of the other games. Find the probability that the stronger team wins the series in exactly i games. Do it for i = 3, 4, 5.