Question

Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 15 purchasers.
The bookstore has 10 new copies and 10 used copies in stock.
If 15 people come in one by one to purchase this text, what is the probability that all 15 will get the type of book they want from current stock?

Answer 1

Answer:

The probability that all 15 will get the type of book they want from current stock is 0.4838.

Step-by-step explanation:

Denote the random variable X as the number of students who want to buy new copy.

The probability of a student wanting to buy a new copy is, P (X) = p = 0.30.

A random sample of n = 15 students is selected.

The random variable X follows a Binomial distribution.

The probability function of a Binomial distribution is:

$P(X=x)={n\choose x}p^{x}(1-p)^{n-x};\ x=0, 1, 2, 3,...$

It is provided that the bookstore has 10 new copies and 10 used copies in stock.

All the 15 students get their desired copy, then this can happen if at most 10 want to buy new copy and at least 5 wants to buy used copy.

Compute the probability of (5 ≤ X ≤ 10) as follows:

P (5 ≤ X ≤ 10) = P (X = 5) + P (X = 6) + P (X = 7) + P (X = 8) + P (X = 9) + P (X = 10)

                     $={15\choose 5}(0.30)^{5}(1-0.30)^{15-5}+{15\choose 6}(0.30)^{6}(1-0.30)^{15-6}\\+{15\choose 7}(0.30)^{7}(1-0.30)^{15-7}+{15\choose 8}(0.30)^{8}(1-0.30)^{15-8}\\+{15\choose 9}(0.30)^{9}(1-0.30)^{15-9}+{15\choose 10}(0.30)^{10}(1-0.30)^{15-10}\\=0.2061+0.1472+0.0811+0.0348+0.0116+0.0030\\=0.4838$

Thus, the probability that all 15 will get the type of book they want from current stock is 0.4838.