# Thread: Conditional Probability and Independence

1. ## Conditional Probability and Independence

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.

2. A)
A=power plant A is running.
B = Power plant B is running.

$\displaystyle P(K=0)=(1-P(A))(1-P(B))= (1-0.4)(1-0.5)=0.3$

$\displaystyle P(K=1) = P(A)(1-P(B))+(1-P(A))P(B)$$\displaystyle =(0.4)(1-0.5)+(1-0.4)(0.5) = 0.2+0.3=0.5$

$\displaystyle P(K=2) = P(A)P(B) = (0.4)(0.5)=0.2$

B)

$\displaystyle E[X]= 1(P(K=2)) +0.6(P(K=1)) +0(P(K=0))$

$\displaystyle E[X]=1(0.2) +0.6(0.5) +0(0.3)=0.2+0.3=0.5$