Math 446 Homework 1
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Math 446 Homework 1
1. A small commuter plane has 30 seats. The probability that any particular passenger will not show up for a flight is 0.05, independent of other passengers. The airline sells 33 tickets for the flight.
a. Calculate the probability that more passengers show up for the flight than there are seats available.
b. Suppose that for each passenger that is “bumped” from the flight, the airline must pay the customer $500. Determine the expected amount the airline will have to pay out. (Hint: Create a new variable that is total payment where $0 is paid when 30 or fewer passengers show up, $500 is paid if $31 passengers show up, etc). The probability for each payment will correspond to the probability of the given number of people who show up.)
2. A company prices its hurricane insurance using the following assumptions:
(i) In any calendar year, there can be at most one hurricane.
(ii) In any calendar year, the probability of a hurricane is 0.05 .
(iii) The number of hurricanes in any calendar year is independent ofthe number of hurricanes in any other calendar year.
Using the company’s assumptions, calculate the probability that there are fewer than 2 hurricanes in a 20-year period .
3. An insurance policy on an electrical device pays a benefit of4000 if the device fails during the first year. The amount of the benefit decreases by 1000 each successive year until it reaches 0 . Ifthe device has not failed by the beginning of any
given year, the probability of failure during that year is 0.4 . (Hint: This is a geometric distribution). What is the expected benefit under this policy?
4. An company tracks their workplace accidents. They determine that each month, the probability ofno workplace accidents is 0.80 and the probability of one or more workplace accidents in a month is 0.2. After the 3rd month with at least one workplace accident, the company will conduct mandatory safety training. Hint: In this scenario, a month with an accident is a “success” and accident free months are “failures”. The distribution for the variable is the total number of accident free months before the third month with an accident.
a. Determine the expected number of months between trainings (i.e. determine the expected value ofthe distribution)
b. Determine the probability that the company will conduct training sometime between and including 11 and 13 months.
5. A company buys a policy to insure its revenue in the event ofmajor snowstorms that shut down business. The policy pays nothing for the first such snowstorm ofthe year and 10,000 for each one thereafter, until the end ofthe year up to a maximum of $50,000. The number ofmajor snowstorms per year that shut down business is assumed to have a Poisson distribution with mean 1.5. What is the probability that the company will pay a claim under this policy during a one-year period? (i.e. What is the probability the number of claims is more than one?). Hint: the amount of payment is not relevant. It is only included to make the problem more realistic.
6. The number of calls received per hour by a single operator at a call center follows a Poisson distribution with a mean of λ = 3.7 calls. Let X be the number of calls received in an hour by an operator. Find P( 2 ≤ X ≤ 4).