A particular type of tennis racket comes in a midsize version and an oversize ve

Question

A particular type of tennis racket comes in a midsize version and an oversize version. 70% of all customers who order this racket from a certain store purchase the oversize version, and 30% purchase the midsize version.

a) Among twelve randomly selected customers who want this type of racket, what is the probability that at least ten want the oversize version?

b) Suppose midsize rackets sell for $109 and oversize rackets sell for $129, and the store currently has 20 of each in stock. What is the expected revenue from the next 12 customers who purchase this racket? What is the standard deviation of revenue from the next 12 customers who purchase this racket?

c) What is the probability that the second oversize racket occurs after the fourth order? (Can you figure out how to do this with both binomial and negative binomial? Which is easier?)

d) What is the probability that the third oversize racket occurs after the ninth order? (Can you figure out how to do this with both binomial and negative binomial? Which is easier?)

e) If 6 of the first 8 orders are for oversize rackets, on average, how many of the first 100 rackets will be for oversize rackets?

f) If 6 of the first 8 orders are for oversize rackets, what is the probability that the 8th oversize racket ordered is the 12th racket ordered overall.

Explanation / Answer

(a)
P(X >= 10) = P(X = 10) + P(X = 11) + P(X = 12)
P(X = 10) = 12C10 * (0.7)^10 * (0.3)^2 = 0.1678
P(X = 11) = 12C11 * (0.7)^11 * (0.3)^1 = 0.0712
P(X = 10) = 12C12 * (0.7)^12 * (0.3)^0 = 0.0138

P(X >= 10) = 0.1678 + 0.0712 + 0.0138 = 0.2528

(b)
Midsize version expected to be sold = 12*0.3 = 3.6 = 4
Oversize version expected to be sold = 12*0.7 = 8.4 = 8

Expected revenue = 4*109 + 8*129 = $1468

(c)
As per given situation, first 3 orders must have exactly one oversize raket order.
hence probability of this is 3C1 * 0.7^1 * 0.3^2 = 0.189

Required probability = 0.189

(d)
As per given situation, first 9 orders must have exactly 2 oversize racket orders.
probability of this is 9C2 * 0.7^2 * 0.3^7 = 0.0039

Required probability = 0.0039

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