Question

A bin of 50 manufactured parts contains 3 defective parts and 47 non-defective parts. A sample of size 6 parts is selected from 50 parts. Selected parts are not replaced. How many different samples are there of size six that contain exactly 2 defective parts? What is the probability that a sample contains exactly 2 defective parts?

Answer 1

Answer:

535,095 different samples of size six that contain exactly 2 defective parts.

0.0337 = 3.37% probability that a sample contains exactly 2 defective parts.

Step-by-step explanation:

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

As the order of the parts is not important, 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.

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

How many different samples are there of size six that contain exactly 2 defective parts?

2 defective from a set of 3, and 4 non-defective from a set of 47. So

$D = C_{3,2}*C_{47,4} = \frac{3!}{2!1!}*\frac{47!}{4!43!} = 535095$

535,095 different samples of size six that contain exactly 2 defective parts.

What is the probability that a sample contains exactly 2 defective parts?

The total number of samples is:

$T = C_{50,6} = \frac{50!}{6!44!} = 15890700$

Then...

$p = \frac{D}{T} = \frac{535095}{15890700} = 0.0337$

0.0337 = 3.37% probability that a sample contains exactly 2 defective parts.