Question

(6) 3. A system comprised of three independent components. The probability of failure for each component is given. #2 .2 #3.3 # 1 .1 A2,A3,A Use our standard Notation: Ali (system operates properly} a) Find the probability that the system operates properly. b) Find the probability that at most two of the components work. (event B) c) Find the probability that the system operates properly if at most two of the components work.

Answer 1

a)

The system works if A1 or (A2 and A3) works. Thus, A = A1 U (A2 $\cap$ A3)

Probability that the system works = P(A) = P(A1 U (A2 $\cap$ A3)) = P(A1) + P(A2 $\cap$ A3) - P(A1 $\cap$ A2 $\cap$ A3)

= (1 - 0.1) + (1 - 0.2) * (1 - 0.3) - (1 - 0.1) * (1 - 0.2) * (1 - 0.3)

= 0.956

b)

Probability that at most two of the components work = P(B)

= 1 - Probability that all three of the components work

= 1 - (1 - 0.1) * (1 - 0.2) * (1 - 0.3)

= 0.496

c)

Probability that the system operates properly if at most two of the components work = P(A | B) = P(A and B) / P(B)

Now,

P(A and B) = Probability that the system operates properly and at most two of the components work = Probability that A1 works and A2, A3 do not work + Probability that A1,A2 works and A3 do not work + Probability that A1, A3 works and A2 do not work + Probability that A2, A3 works and A1 do not work

= (1 -0.1) * 0.2 * 0.3 + (1 - 0.1) * (1 - 0.2) * 0.3 + (1 - 0.1) * 0.2 * (1 - 0.3) + 0.1 * (1 - 0.2) * (1 - 0.3)

= 0.452

Probability that the system operates properly if at most two of the components work = P(A | B) = P(A and B) / P(B)

= 0.452 / 0.496

= 0.9112903