Question

5. A pumping system consists of two identical pumps connected in parallel, so that if either pump fails, the system still operates. The probability that a pump will fail is during the period for which it was designed is 10%

a. What is the probability that the system works throughout the period for which it was designed?

b. How many pumps in parallel are needed if we want the probability that the system will work throughout its useful life is not less than 0.9999?

c. Let us assume that although the pumps are initially identical, if one of them fails then the additional effort required to make the second pump causes the probability that it will fail to double (the initial probability of failure is unknown, it is NOT 10%). If we know that in 7% of the systems at least one pump fails during the period for which they were designed, but that only 1% of the systems fail completely, Calculate the initial probability of failure for each pump.

Answer 1

Question 5

P(A single pump will fail) = 0.10

(a) P(System works throughout the period) = 1- P(Both pump fails) = 1- (1-0.9) * (1-0.9) = 0.99

(b) Here let say n pumps are needed

so,

0.9999 = 1 - (1- 0.9)ⁿ

0.1ⁿ = 1 - 0.9999 = 0.0001

n = 4

(c) Let say initial probability for failure of each pump = p1 and p2

so here

P(At least one pump fail) = p₁ + p₂ - 4p₁p₂ = 0.07

P(Both system fail) = p₁* 2p₂ + p₂ * 2p₁= 4p₁p₂ = 0.01

p₁p₂ = 0.0025

p₁ + p₂ = 0.06

(0.06 - p₂) * p₂ = 0.0025

0.06p₂ - p₂² - 0.0025 = 0

p₂ = 0.0466 and p₁ = 0.0134