Question

An important problem encountered by personnel directors and others faced with the selection of the best in a finite set of elements is exemplified by the following scenario. From a group of 20 engineers, including 5 Ph.D. engineers, 10 are randomly selected for employment. What is the probability that the 10 selected include include all the best 5 engineers in the group of 20?

Answer 1

Answer:

1.63% probability that the 10 selected include include all the best 5 engineers in the group of 20

Step-by-step explanation:

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

The order in which the engineers are selected is not important, so the combinations formula is used 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.

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

Desired outcomes:

10 engineers selected from a set of 20.

5 best, from a set of 5.

Other 5, from a set of 20-5 = 15.

$D = C_{5,5}*C_{15,5} = \frac{5!}{5!(5-5)!}*\frac{15!}{5!(15-5)!} = 3003$

Totao outcomes:

10 engineers selected from a set of 20.

$T = C_{20,10} = \frac{20!}{10!(20-10)!} = 184756$

Probability:

$p = \frac{D}{T} = \frac{3003}{184756} = 0.0163$

1.63% probability that the 10 selected include include all the best 5 engineers in the group of 20