Question

Two power plants (A and B) operate in parallel to power a city. The demand for power is subject to considerable fluctuation. Each unit has the capacity to 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. The probability both units fail is 0.02. That is,

A = event unit A fails

B = event unit b fails

P(A) = P(B) = 0.10 P(AB) = 0.02

a) Determine P(A|B) and P(B|A)

b) Use result from a) and determine the conditional probability that when there is failure, only one of the two units fail

c) Use the result from b) to determine the probability the city will have of full power, when there is failure of one unit.

Answer 1

a)

P(A|B) =P(AB)/P(B) =0.02/0.1 =0.2

P(B|A) =P(AB)/P(A) =0.02/0.1 =0.2

b) P(at least one failure) =P(A u B)=P(A)+P(B)-P(A n B) =0.1+0.1-0.02=0.18

P(only one of two units fail)=P(A)+P(B)-2P(A n B) =0.1+0.1-2*0.02=0.16

conditional probability that when there is failure, only one of the two units fail

=P(only one fail|at least one fail)=0.16/0.18=0.8889

c)

P( city will have of full power, when there is failure of one unit) =one unit works and fullfill city's power requirement=0.75*0.8889=0.6667