Question

A library patron is trying to find books for her family to enjoy. Suppose that when the patron looks
at a book, there is a 75% chance that she will check the book out. The patron has decided to check
out 8 books. Assume that whether she decides to check out a book is independent of the other
books she looks at. Let X be the number of books that she decides not to check out (and has to
reshelve) before she gets to the eighth book to check out.
a) (1) Note that X has a negative binomial distribution. What are the values of r and p?
b) (2) What is the probability that the library patron will have exactly 2 books she decides not to
check out (before she gets to the eighth book to check out)?
c) (3) What is the probability that she will have 2 or fewer books that she decides not to check out
(before she gets to the eighth book to check out)?
d) (1) How many books should the library patron expect to look at but not check out (on average,
before she gets to the eighth book to check out); that is, determine the mean of X.
e) (1) Determine the standard deviation in X.

Answer 1

a)

Here we need 8th successes and the probability of one success is 0.75 so we have

r = 8, p=0.75

b)

Since X shows the number of books that she decides not to check out (and has to reshelve) before she gets to the eighth book to check out is

The  probability that the library patron will have exactly 2 books she decides not to check out (before she gets to the eighth book to check out) is

c)

The probability that she will have 2 or fewer books that she decides not to check out (before she gets to the eighth book to check out) is

d)

e)