Question

A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the remaining pump is now more likely to fail than was originally the case. That is, r = P(#2 fails | #1 fails) > P(#2 fails) = q. If at least one pump fails by the end of the pump design life in 10% of all systems and both pumps fail during that period in only 4%, what is the probability that pump #1 will fail during the pump design life?

Answer 1

0.07 or 7%

Explanation:

Since both pumps are identical, it is reasonable to assume that they have the same probability of failing, that is P(#1 fails)=P(#2 fails).

The probability that at least one pump fails is:

`P(fail>0) = P(\#1 fails)+P(\#2 fails)- P(\#1\ and\ \#2 fail)\\0.10 = P(\#1 fails)+P(\#2 fails) - 0.04\\P(\#1 fails)+P(\#2 fails) = 0.14`

Since both pumps fail at the same rate, the probability that pump #1 will fail during the pump design life is:

`2*P(\#1 fails) = 0.14\\P(\#1 fails) =0.07=7\%`

The probability is 0.07 or 7%.