32. Suppose that, at the local record store, there are 2 customers in the checko

Question

32. Suppose that, at the local record store, there are 2 customers in the checkout line. Each customer buys a number of records given by a random variable X with the following probability mass function Page 5 (x 0.5 0.2 0.1 0.2 Further, the number of records purchased by any customer is independent of the number (a) What is the probability that (between the two customers currently in line) exactly (b) Now suppose there are 3 customers in line (and they independently buy records of records bought by any other customer 4 records are sold? with this same probability distribution). What is the probability that (between these three customers) 5 or fewer records are sold?

Explanation / Answer

(a)

Each customer can buy 1, 2, 3 ro 4 records. Let X1 shows the number of records purchased by first customer and X2 shows the number of records purchased by second customer. Let Y = X1+X2

Following table shows all possible values of X1, X2 and Y and corresponding probabilities:

Following table shows the probability distribution of Y:

So the probability that exactly 4 records are sold is

P(X=4) = 0.14

(b)

The three customers can buy 5 or fewer records if : all three buy one record, two buy 1 record and third buy 2 or 3, one buy one record and 2 buy 2 records.

P(all buy one record) = 0.5*0.5*0.5 = 0.125

P(two buy one record and one buy 2 records) = C(3,1)*0.5*0.5*0.2 = 0.15

Here C(3,1) shows the number of ways of selecting one person out of three for buying 2 records.

P(two buy one record and one buy 3 records) = C(3,1)*0.5*0.5*0.1 = 0.075

Here C(3,1) shows the number of ways of selecting one person out of three for buying 3 records.

P(one buy one record and two buy 2 records) = C(3,1)*0.5*0.2*0.2 = 0.06

Here C(3,1) shows the number of ways of selecting one person out of three for buying 1 records.

Hence, required probability is

P(5 or fewer records) = 0.06+ 0.15 +0.075 = 0.285

X1 X2 Y=X1+X2 P(X1=x1) P(X2=x2) P(Y=y)=P(X1=x1)P(X2=x2) 1 1 2 0.5 0.5 0.25 1 2 3 0.5 0.2 0.1 1 3 4 0.5 0.1 0.05 1 4 5 0.5 0.2 0.1 2 1 3 0.2 0.5 0.1 2 2 4 0.2 0.2 0.04 2 3 5 0.2 0.1 0.02 2 4 6 0.2 0.2 0.04 3 1 4 0.1 0.5 0.05 3 2 5 0.1 0.2 0.02 3 3 6 0.1 0.1 0.01 3 4 7 0.1 0.2 0.02 4 1 5 0.2 0.5 0.1 4 2 6 0.2 0.2 0.04 4 3 7 0.2 0.1 0.02 4 4 8 0.2 0.2 0.04 Total 1

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