Question

A study found that 25% of car owners had their cars washed professionally rather than do

it themselves. If 18 car owners are randomly selected, find the probability that at most two
people have their cars washed professionally.​

Answer 1

13.52% probability that at most two people have their cars washed professionally.​

Explanation:

For each car owner, there are only two possible outcomes. Either they have their car washed professionally, or they do not. The probability of a car owner having their car washed professionally is independent of other car owners. 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.

25% of car owners had their cars washed professionally

This means that
`p = 0.25`

18 car owners

This means that
`n = 18`

Probability that at most two people have their cars washed professionally.​

`P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)`

In which

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

`P(X = 0) = C_(18,0).(0.25)^(0).(0.75)^(18) = 0.0056`

`P(X = 1) = C_(18,1).(0.25)^(1).(0.75)^(17) = 0.0338`

`P(X = 2) = C_(18,2).(0.25)^(2).(0.75)^(16) = 0.0958`

`P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0056 + 0.0338 + 0.0958 = 0.1352`

13.52% probability that at most two people have their cars washed professionally.​