Question

An important part of the customer-service responsibilities of a telephone company relates to the speed with which trouble in residential service can be repaired. Suppose that past data indicates that the probability is 0.9 that troubles in residential can be repaired in the same day. For the first five troubles reported on a given day. What is the probability that all troubles will be repaired in the same day

Answer 1

59.05% probability that all troubles will be repaired in the same day

Explanation:

For each trouble, there are only two possible outcomes. Either it can be repaired on the same day, or it cannot. The probability of a trouble being repaired on the same day is independent of other troubles. 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.

Suppose that past data indicates that the probability is 0.9 that troubles in residential can be repaired in the same day

This means that
`p = 0.9`

For the first five troubles reported on a given day. What is the probability that all troubles will be repaired in the same day

This is P(X = 5) when n = 5. Then

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

`P(X = 5) = C_(5,5).(0.9)^(5).(0.1)^(0) = 0.5905`

59.05% probability that all troubles will be repaired in the same day