Question

A state trooper is setting up a speed trap on the turnpike. he knows that approximately 75% of drivers go at least 5 miles over the posted speed limit. what is the probability that in a sample of 10 cars that pass him, exactly 5 are speeding?

Answer 1

There is a 5.84% probability that in a sample of 10 cars that pass him, exactly 5 are speeding.

Explanation:

For each car that passes the trap, there are only two possible outcomes. Either they are speeding, or they are not. This means that 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 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 we have that:

`n = 10, p = 0.75`

What is the probability that in a sample of 10 cars that pass him, exactly 5 are speeding?

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

`P(X = 5) = C_(10,5).(0.75)^(5).(0.25)^(5) = 0.0584`

There is a 5.84% probability that in a sample of 10 cars that pass him, exactly 5 are speeding.