Question

The Copy Shop has made 20 copies of a document for you. Since the defective rate is 0.1, you think there may be some defective copies in your order, so you leaf through the first ten (which are a randomly chosen subset).

If there are 2 defective copies among the 20, what is the probability that you will encounter neither of the defective copies among the 10 you examine?


X = number of copies with a defect


a. X ~ binomial


b. X ~ negative binomial


c. X ~ hypergeometric


d. X ~ Poisson

Answer 1

Binomial

There is a 34.87% probability that you will encounter neither of the defective copies among the 10 you examine.

Explanation:

For each copy of the document, there are only two possible outcomes. Either it is defective, or it is not. This means that we can solve this problem using the binomial probability distribution.

Binomial probability distribution:

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinatios of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

In this problem

Of the 20 copies, 2 are defective, so
`p = (2)/(20) = 0.1`.

What is the probability that you will encounter neither of the defective copies among the 10 you examine?

This is P(X = 0) when
`n = 10`.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(10,0).(0.1)^(0).(0.9)^(10) = 0.3487`

There is a 34.87% probability that you will encounter neither of the defective copies among the 10 you examine.