Question

Armco, a manufacturer of traffic light systems, found 95% of the newly developed systems lasted 3 years before failing to change signals properly. If a city purchased four of these systems, what is the probability all four systems would operate properly for at least 3 years

Answer 1

81.45% probability all four systems would operate properly for at least 3 years

Explanation:

For each traffic light, there are only two possible outcomes. Either they work for at least 3 years, or they do not. The probability of a light working at least 3 years is independent of other lights. 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.

95% of the newly developed systems lasted 3 years

This means that
`p = 0.95`

The city purchases four systems

This means that
`n = 4`

What is the probability all four systems would operate properly for at least 3 years

This is P(X = 4).

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

`P(X = 4) = C_(4,4).(0.95)^(4).(0.05)^(0) = 0.8145`

81.45% probability all four systems would operate properly for at least 3 years