Suppose that 27% of all students who have to buy a text for a particular course want a new copy...

Question:

Suppose that 27% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 73% want a used copy. Consider randomly selecting 25 purchasers.

The bookstore has 15 new copies and 15 used copies in stock. If 25 people come in one by one to purchase this text, what is the probability that all 25 will get the type of book they want from current stock? (Hint: Let X=X= the number who want a new copy. For what values of XX will all 25 get what they want?)

Connecting with Binomial Theorem

In this problem we are given the notion that we are working with Binomial distribution. This is because we are given two important probabilities, the probability of success, p = 0.27 , and the probability of failure, (1-p) = 0.73. Thus, in this problem, we will make use of the following equation:

{eq}P(a \lt x \lt b) = \sum_{x=a}^{b} {\binom{n}{x} (p)^x (1-p)^{n-x}} {/eq}

From the information in the problem we can note that everyone will get the type of book they want if at most, only 15 want used books and 15 wants new books from the bookstore. This can be determined as {eq}P(25-x \gt 15) = P(X \gt10) {/eq} or as, {eq}P(X \lt 15) {/eq}

Thus, when we combine the two ideas, we get

{eq}P(10 \lt x \lt 15) = \sum_{x=10}^{15} {\binom{25}{x} (0.27)^x (0.73)^{25-x} }= 0.11 {/eq}