Question

1. There are several scenarios described below. For each of them, do the following (note: R.V. means random variable) (1) Define the R.V.--- that means something like, “Let X be the number of people who....." (2) Define the distribution and parameter(s) of the R.V. (3) Give the support of the R.V. (4) Write the probability statement related to the information being sought. Do not calculate the probability. a) The Bureau of Motor Vehicles in Michigan states that 4 out of 5 people pass the written driver's test. Bob goes to the BMV to take the written test. What is the probability that Bob fails the test? b) The store manager at BigMart knows that after 5 pm, on average there are 20 customers leaving the store every 30 minutes. What is the probability that 5 customers leave Walmart between 6:10 and 6:20 pm? c) A BigMart store is going to hire 5 new workers. It has a pool of 15 applicants (9 male, 6 female) for these 5 jobs. The BigMart manager wants to determine the probability that 2 of new hires are females. d) A gardener is inspecting her the fall flowers in her garden. She notices, on average, 4 bugs on a flower. She randomly picks one flower from her garden. She wants to know the probability that the flower she picked has at least one bug on it. e) Harry is a salesman at Gippo's Appliance World. He needs to sell 4 appliances today. Usually, Harry is successful 37% of the time that he talks to customers. What is the probability that he will meet his quota of 4 sales before he talks to his 10th customer of the day? f) A certain radio station's phone lines are busy approximately 95% of the time when trying to call during a contest. We are interested in finding the probability that the 4th time you call is the 1st time you get through during a contest.

Answer 1

b) X = No.of customers leaving the store after 5 pm, every 30 mins X is Poisson RV with X = 20 Support = {0,1,2,3,.......};; info being sought : Let Y : No.of customers leaving the store after 5 pm, every 10 mins, ly = 20 * 10/30 = 6.666 We want P(Y = 5)

c) X = No.of female hires in the group of 5 new hires. X is Hyper geometric RV with population size N = 15, sample size n = 5; K = 6 females : no.of successes k= 2: Support = {0, 1, 2, 3, 4,5};: info being sought : Let We want P(X=2)

$d) \ X = No.of\ bugs\ on\ randomly\ chosen\ flower. \\ \ X\ is\ Poisson\ RV\ with\ \lambda = 4; \\ \\ Support\ = \{0,1,2,3,4,5,.....\} ; ; \ info\ being\ sought : \\ We\ want \ P(X\ge 1) \\ \\$

$e) \ X = no.\ of\ customers\ Harry\ talks\ to\ until\ 4\ sales. \\ \ X\ is\ Negative\ binomial\ RV\ with\ k = 4; p=0.37 \\ \\ Support\ = \{4,5,6,....\} ; ; \ info\ being\ sought : \\ We\ want \ P(X\le 9) \\ \\$

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