A certain market has both an express checkout line and a superexpress checkout l

Question

A certain market has both an express checkout line and a superexpress checkout line. Let X denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number uf customers in line at Lhe super express checkout at the same time. Suppose the joint pm of X1 and X2 is as given in the accompanying table. X2 0 0.09 0.07 0.04 0.00 10.06 0.15 0.05 0.04 x1 2 0.05 0.03 0.10 0.06 3 0.00 0.04 0.04 0.07 4 0.00 0.02 0.05 0.04 (a) What is P(X - 1, X2 -1), that is, the probability that there is exactly one customer In cach linc? PXi-1, x21) (b) what is P(X1-X2), that is, the probability that the numbers of customers in the two lines are identical? (c) Let denote the event that there are at least two more customers in one line than in the other line. Express in terms of X1 and X2. O^-(x, z 2 + x, ux, z 2 +%) Calculate the probabllity of thls event (d) What is the probability that the total number ot customers in the two lines is exactly tour? At least four? P(exactly four) - P(at least four)

Explanation / Answer

a) P(X1 = 1 and X2 = 1) = 0.15

b) P(X1 = X2) = P(0,0) + P(1,1) + P(2,2) + P(3,3) = 0.09 + 0.15 + 0.1 + 0.07 = 0.41

c) Option-C)

P(A) = P(0,2) + P(0,3) + P(1,3) + P(2,0) + P(3,0) + P(3,1) + P(4,0) + P(4,1) + P(4,2)

        = 0.04 + 0 + 0.04 + 0.05 + 0 + 0.04 + 0 + 0.02 + 0.05

        = 0.24

d) P(exactly 4) = P(1,3) + P(2,2) + P(3,1) + P(4,0)

                        = 0.04 + 0.1 + 0.04

                        = 0.18

P(at least 4) = P(exactly 4) + P(exactly 5) + P(exactly 6) + P(exactly 7)

                     = 0.18 + (P(2,3) + P(3,2) + P(4,1)) + (P(3,3) + P(4,2)) + P(4,3)

                     = 0.18 + (0.06 + 0.04 + 0.02) + (0.07 + 0.05) + 0.04

                     = 0.46

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