Question

In a class of 100 students, 25 have hardcover and 75 students have paperback textbooks. If you randomly choose 15 students in this class, find the probability that 3 of them have hardcover texts.

Answer 1

Answer:

23.72% probability that 3 of them have hardcover texts.

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

The order in which the students are chosen is not important. So we use the combinations formula to solve this question.

Combinations formula:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

If you randomly choose 15 students in this class, find the probability that 3 of them have hardcover texts.

Desired outcomes:

3 with hardcover texts, from a set of 25.

15-3 = 12 with paperback texts, from a set of 75. So

$D = C_{25,3}*C_{75,12} = \frac{25!}{3!(22)!}*\frac{75!}{12!(63)!} = 6 \times 10^{16}$

Total outcomes:

15 students from a set of 100. So

$T = C_{100,15} = \frac{100!}{15!85!} = 2.53 \times 10^{17}$

Probability:

$P = \frac{D}{T} = \frac{6 \times 10^{16}}{2.53 \times 10^{17}} = 0.2372$

23.72% probability that 3 of them have hardcover texts.