(20pts) Consider a lottery in which each ticket consists of 4 unique numbers tak

Question

(20pts) Consider a lottery in which each ticket consists of 4 unique numbers taken from the set N = {1, 2, ,20). The numbers in a ticket are always sorted in increasing order; for example, 10-16-17-19 and 1-2-3-4 are valid tickets, but 7-10-11-8 is not. Your friend Hurley just bought ticket 4-8-15-16. When the lottery plays, 4 numbers are drawn from N with equal probability and without replacement. An outcome of this experiment corresponds to each possible set of 4 numbers drawn. Let X be the random variable that represents how many numbers from Hurley's ticket are drawn. For example, X(1-16-17-18) 1 and X(4-8-15-19) 3. Answer the following questions: a) (4pts) How many possible tickets can be constructed in this lottery? b) (4pts) How many outcomes are in {X = 3)? Give some examples and justify your answer, what is the probability of (X = 3)? C) (4pts) What is the probability that {X 2 2]? d) (Apts) What are the values of E[X] and var(X)? e) (4pts) Suppose the lottery payouts as follows: $1 for tickets that have 2 numbers matching the numbers drawn, S10 for tickets that have 3 numbers matching the 4 numbers drawn, and $100 for tickets that match all 4 of the numbers drawn. What is Hurley's expected payout?

Explanation / Answer

a) Number of tickets = 20C4=20!/[16!*4!]= 4845

b) Outcomes in X=3 are 4- Any 1 number can be incorrect while remaining 3 correct

P(X=3)=4/4845

c)P(X<=2) = (6+4+1)/4845

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