# Independent Probability — Chance of failure

I've got the following problem that I keep getting the wrong answer and looking to see if someone can help me understand how to get the right answer:

"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)?"

Naturally the first thing I thought to do was .1 * .05 = .005 however the correct answer is .0059.

I've then noticed the following but still haven't managed to get the answer.

P(B) = 0.05 <--- New one fails probability

P(old one fails AND new one does not fail) = 0.1 = P(A | B )

I feel like I'm on the right track but don't really see what to do to get .0059.

## 2 Answers

Hint: If $$p$$ is the probability that the older pump fails and $$q$$ is the probability that the newer pump fails then based on the probabilities that one fails and the other does not, you have $$p(1-q)=0.1$$ $$(1-p)q=0.05$$ while you are trying to find the probability that both of them fail, which is $$pq.$$

Approximately, $$0.1059 \times (1-0.0559) \approx 0.1$$, $$(1-0.1059) \times 0.0559 \approx 0.05$$, and $$0.1059 \times 0.0559 \approx 0.0059$$, so your suggested answer is close to the correct answer.

• Are you saying to do .1 + .05 + [(.05)(.1)] = .155? I'm not quite sure I understand what you mean by pq – Valrok Feb 17 '14 at 23:33
• @Henry, good start but surely the question requires the probability that both pumps fail, which in your notation is $pq$. – David Feb 17 '14 at 23:33
• @David: Thank you: you are correct - I will edit – Henry Feb 17 '14 at 23:39
• Why is pq the probability that either of them fail? Why would either failing be a smaller chance than each one separately failing? – Aaron Franke Sep 27 '18 at 21:13
• @AaronFranke $pq \approx 0.0059$ is the probability both fail; while $p(1-q)+(1-p)q=0.15$ is the probability exactly one fails and $p+q-pq \approx 0.1559$ is the probability that at least one fails – Henry Sep 27 '18 at 21:25

Let $x$ be the probability that they both fail. Then the probability the older one fails is $0.1+x$, and the probability the newer one fails is $0.05+x$. Thus by independence the probability that they both fail is $(0.1+x)(0.05+x)$. But this is equal to $x$. We obtain a quadratic equation in $x$. Solve.

Remark: The quadratic equation is $x^2-0.85x+0.005=0$. Note that this has two solutions, one of which is approximately $0.0059236$. The other solution is approximately $0.844$.