Answer:
a) There is a 10.24% probability that both generators fail during a power outage.
b) There is an 89.76% probability of having a working generator in the event of a power outage, which is not high enough for the hospital.
Step-by-step explanation:
For each emergency backup generator, there are only two possible outcomes. Either they work correctly, or they fail. So 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.
In which is the number of different combinations of x objects from a set of n elements, given by the following formula.
And p is the probability of X happening.
In this problem we have that:
Assume that emergency backup generators fail 32% of the times when they are needed. So they work correctly 100-32 = 68% of the time. So
There are two generators, so
a. Find the probability that both generators fail during a power outage
This is
So
There is a 10.24% probability that both generators fail during a power outage.
b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital? Assume the hospital needs both generators to fail less than 1% of the time when needed.
Either there are no working generators, or there is at least one working generator. The sum of the probabilities of these events is decimal 1. So
We want . So
There is an 89.76% probability of having a working generator in the event of a power outage, which is not high enough for the hospital.
To be high enough for the hospital, this probability should be at least of 99%.