Two pumps connected in parallel fail independently of one another on any given day. The probability that only the older pump will fail is .10, and the probability that only the newer pump will fail is .05. What is the probability that the pumping system will fail on any given day (which happens if both pumps fail)?

Answer : Step 1 : Given Two pumps connected in parallel fail independently of one another on any given day. The probability that only the older pump will fail is .10, and the probability that only the newer pump will fail is .05. We need to find the probability that the pumping system will fail on any given day (which happens if both pumps fail). We know that P(O)=0.10 and P(N)=0.05 Then, P(O )= P(O) - P(ON) and P(N ) = P(N) - P(OnN). where O is the event that the older pump fails and N is the event that the newer pump fails. So P(O) = P(O) + P(O N) and P(N) = P(N) + P(O N). P(O) = P(O) + P(O) P(N) P(O) = 0.10 + (0.10 × 0.05) P(O) = 0.05 + 0.005 P(O) = 0.105 And P(N) = P(N) + P(O) P(N) P(N) = 0.05 + (0.10 × 0.05) P(N) = 0.05 + 0.005 P(N) = 0.055 As the events O and N are independent. So P(O)×P(N) = 0.105 × 0.055 = 0.00577