Suppose electric power is supplied from two independent sources which work with probabilities 0.4, 0.5, respectively. if both sources are providing power enough power will be available with probability 1. if exactly one the them works there will be enough power with probability 0.6. of course, if none of them works the probability that there will be sufficient supply is 0. a) what are the probabilities that exactly k sources work for k=0,1,2? b) compute the probability that enough power will be available.

The answers to the questions are as follows

a) k = 0, P = 0.3

k = 1, P = 0.5

k = 0, P = 0.2

b) The probability that enough power will be available is 0.5.

Explanation:

To solve the question we write the parameters as follows

Probability that the first power source works = P (A) = 0.4

Probability that the second power source works = P (B) = 0.5

When both sources are supplying power we have the probability = 1

If non of them is producing the probability = 0

a) The probability that exactly k sources work for k=0,1,2 is given by

For k = 0, probability = (1 - P (A)) * (1 - P (B)) = 0.6 * 0.5 = 0.3

Therefore the probabilities that exactly 0 source work = 0.3

for k = 1 we have the probability = P (A) (1-P (B)) + P (B) (1-P (A)

= 0.4 (1-0.5) + 0.5 (1-0.4) = 0.2 + 0.3 = 0.5

The probabilities that exactly 1 source work = 0.5

for k = 2 we have the probability given by = P (A) * P (B) = 0.4 * 0.5 = 0.2

Therefore the probability that exactly 2 sources work = 0.2

b) The probability that enough power will be available is

0 * P (k = 0) + 0.6 * P (k = 1) + 1 * P (k = 2)

0 * 0.2 + 0.6 * 0.5 + 1 * 0.2 = 0.5

The probability that enough power will be available is 0.5.