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

Answer:

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

Step-by-step 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.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

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