Cupons Collector Problem - Coupon Collection Box - Bus Stop - Simulation - Mathematics Assessment Answer

Cupons Collector Problem - Bus Stop - Simulation - Mathematics Assessment Answer

Assessment Task:

Problem 1

This is called the "Cupon Collector's Problem," 10 different coupons are distributed into cereal boxes, on the per box, so that any randomly selected box is equally likely to have any of the 10 coupons inside. Write a program to simulate the process of buying cereal boxes until all 10 distinct coupons have been collected. For each run, keep track of how many cereal boxes you purchased to collect the complete set of coupons. Then use your program to answer the following questions. 
(a) What is the probability you collect all 10 coupons with just 10 cereal boxes? 
(b) Use counting techniques to determine the exact probability in (a). Hint: Relate this to the Birthday Problem.

(c) What is the probability you require more than 20 boxes to collect all 10 coupons?

(d) What is the probability of collecting all 10 coupons in fewer than average boxes (i.e., less than 29.3)? (e) Let X = the number of cereal boxes purchased in order to obtain all 10 coupons. Use a simulation program to estimate E(X) and SD(X). (f) Repeat

(e) with 20 coupons instead of 10. Does it appear to take roughly twice as long to collect 20 coupons as 10? More than twice as long? Less? 

Problem 3.

You show up at a bus stop to wait for a bus that comes by once per hour. You do not know what time the bus came by last. Let X = the arrival time of the next bus measured in hours, X ,,, U(0,1). (a) Suppose you go to this bus stop once a day for work 5 days a week. Simulate a month (4 weeks) of the amount of time you waited for the bus. 
(b) From your simulation, give a histogram for the amount of time in a month that you spent waiting for the bus.

(c) From your simulation, give the mean and standard deviation.

(d) Repeat (a) - (c) for a year. 
(e) Determine the expected value and standard deviation for the amount of time waiting for the bus based on X ,,, U(0, 1). Compare this to the means and standard deviations derived before. 

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