Question

Approximately 26% of all pipework failures in chemical plants are caused by operator error. What is the probability that out of the next 30 pipework failures, at least 14 are due to operator error

Answer 1

1.17% probability that out of the next 30 pipework failures, at least 14 are due to operator error.

Explanation:

For each pipework failure in chemical plants, there are only two possible outcomes. EIther it is caused by operator error, or it is not. For each failure, the probability of it being caused by operator failure is independent from other failures. So we use the binomial probability distribution to solve this problem.

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.

Approximately 26% of all pipework failures in chemical plants are caused by operator error.

This means that
`p = 0.26`

What is the probability that out of the next 30 pipework failures, at least 14 are due to operator error

This is
`P(X \geq 14)` when
`n = 30`

Using a binomial probability calculator, we have that

`P(X \geq 14) = 0.0117`

1.17% probability that out of the next 30 pipework failures, at least 14 are due to operator error.