Question

If we inspect photocopy paper by randomly drawing 5 sheets without replacement from every pack of 500, what is the probability of getting 5 clean sheets although 0.4% of the sheets contain spots?

Answer 1

Answer:

98.01% probability of getting 5 clean sheets

Step-by-step explanation:

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

In this problem, the order that the sheets are chosen is not important, so we use the combinations formula to solve this problem.

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.

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

0.4% of the sheets contain spots

Of 500, that is 0.004*500 = 2.

What is the probability of getting 5 clean sheets although 0.4% of the sheets contain spots?

Desired outcomes

2 are defective, so 5 sheets from a set of 500 - 2 = 498.

$D = C_{498,5} = \frac{498!}{5!493!} = 250160254344$

Total outcomes

5 sheets from a set of 500.

$T = C_{500,5} = \frac{500!}{5!495!} = 255244687600$

Probability:

$P = \frac{D}{T} = \frac{250160254344}{255244687600} = 0.9801$

98.01% probability of getting 5 clean sheets