# Two power generating units A and B operate in parallel to supply the power requirement of a small city. If there is failure in the power generation what is the probability that the city will have its supply of full power?

## The demand for power is subject to considerable fluctuation, and it is known that each unit has a capacity so that it can supply the city's full power requirement 75% of the time in case the other unit fails. The probability of failure of each unit is 0.10, whereas the probability that both units fail is 0.02.

Sep 22, 2016

15%

#### Explanation:

The probability of failure given is
$p \left(A = f a i l\right) = .10$
$p \left(B = f a i l\right) = .10$
$p \left(A = f a i l \wedge B = f a i l\right) = .02$

To determine the amount of power that they may have we need to account for the amount of power that can be supplied at any given time.

If just one fails then the value of $.75$ is possible. If both are not working then there is no power.

We can use expectation here where we use the definition for discrete events as
$E \left(X\right) = \sum X p \left(X\right)$
In this case we take all the possible values it can take on for each outcome. That is to say what we expect to see for failures is
$E \left(P o w e r | F a i l u r e\right) = .10 \cdot .75 + .10 \cdot .75 + .02 \cdot 0 = .15$

So if there is a failure there is a 15% probability that there will be full power at any given time.