5. Every time Sherry buys a book it is a math book with probability 0.3 and a no

Question

5. Every time Sherry buys a book it is a math book with probability 0.3 and a novel with probability 0.7. Sherry keeps her math books in a bookcase that can hold 50 math books and keeps her novels in a different bookcase that can hold 100 novels. What is the probability that she fills up her math bookcase before she fills up her novels bookcase, assuming they are initially empty? Assume Sherry continues her book buying patterns even after one or both of her bookcases gets filled. Hint: The event that 50 math books are purchased before 100 novels are purchased is the same as the event that in the first 149 purchases, at least 50 of them are math books. Use indicator random variables and the central limit theorem to approximate the desired probability.

Explanation / Answer

Solution:

Given information
n = 149, p = 0.3
? = np = 149 * 0.3 = 44.7
? = sqrt(np(1-p)) = sqrt(149*(0.3(1-0.3)) = 5.5937

Since we are approximating a discrete distribution by continuous normal distri-bution, values between 49.5 and 50.5 both approximate to 50. Thus, "greater than or equal to 50 " corresponds to continuous normal distribution with X > 49.5 after continuity correction.

z = 49.5 - 44.7/5.5937
= 0.8581

P(X > 49.5) = P(Z > 49.5 - 44.7/5.5937)
= P(Z > 0.8581)
= 1- P(Z < 0.8581)
= 0.1949
P(X ? 50) = 0.1949

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