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Probability Challenges: Red Balls, Insurance Claims, and Loss Distributions

This collection presents a series of probability problems involving red and white balls, claims data in insurance, and loss distributions for various events. The problems cover drawing without replacement, independent random variables, and continuous distributions. Each section provides a detailed probability question followed by its calculated answer, such as the probability of obtaining a certain combination of balls and the variance of claims received over a defined period. Ideal for students and professionals looking to enhance their understanding of probability theory and its applications.

darius-mccall
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Probability Challenges: Red Balls, Insurance Claims, and Loss Distributions

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  1. Practice Problems Actex 3, 4, 5

  2. Section 3 -- #3 A box contains 4 red balls and 6 white balls. A sample of size 3 is drawn without replacement from the box. What is the probability of obtaining 1 red ball and 2 white balls, given that at least 2 of the balls in the sample are white? Answer: 0.75

  3. Section 3 -- #6 An insurance company determines that N, the number of claims received in a week, is a random variable with P(N=n) = 1/2n+1, where n => 0. The company also determines that the number of claims received in a given week is independent of the number of claims received in any other week. Determine the probability that exactly seven claims will be received during a given two-week period. Answer: 1/64

  4. Section 4 -- #5 In a small metropolitan area, annual losses due to storm, fire, and theft are independently distributed random variables. The pdf’s are: Storm: f(x) = e-x Fire: f(x) = (2/3)e-2x/3 Theft: f(x) = (5/12)e-5x/12 Determine the probability that the maximum of these losses exceeds 3. Answer: 0.414

  5. Section 4 -- #10 • An insurance company insures a large number of homes. The insured value, X, of a randomly selected home is assumed to follow a distribution with density function, f(x)=3x-4 for x>1 and equal to zero otherwise. Given that a randomly selected home is insured for at least 1.5, what is the probability that it is insured for less than 2? Answer: 0.578

  6. Section 4 -- #16 The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function f, where f(x) is proportional to (10+x)-2. Calculate the probability that the lifetime of the machine part is less than 6. Answer: 0.46875

  7. Section 5 -- #7 A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e. everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car? Answer: 374.4

  8. Section 5 -- #23 A random variable has the cumulative distribution function: F(x)= 0 for x < 1 (x2-2x+2)/2 for 1 <= x < 2 1 for x => 2 Calculate the variance of X. Answer: 5/36

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