Question

A (very bad) factory produces plastic bags such that 30% of the plastic bags produced are defective and 70% are good. A sample of 5 plastic bags is selected at random. (a) What is the probability that all 5 bags selected are defective?

Answer 1

0.24% probability that all 5 bags selected are defective

Explanation:

For each bag, there are only two possible outcomes. Either they are defective, or they are not. The probability of a bag being defective is independent from other bags. So we use the binomial probability distribution to solve this question.

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 combinations 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.

30% of the plastic bags produced are defective

This means that
`p = 0.3`

A sample of 5 plastic bags is selected at random

This means that
`n = 5`

What is the probability that all 5 bags selected are defective?

This is P(X = 5). So

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

`P(X = 5) = C_(5,5).(0.3)^(5).(0.7)^(0) = 0.0024`

0.24% probability that all 5 bags selected are defective