0
$\begingroup$

I need to find the probability that component 1 works given that the system works. All components fail independently. I have attached the network here:

The answer must be in terms of $p_1, p_2, p_3, p_4$.

So far I have that: $$\mathbb{P}(\text{Component }i\text{ works}) = p_i.$$ We know that the question is $\mathbb{P}(W_1 \mid S)$ where $W_1$ is the event that component 1 works and $S$ is the event that the system works.

$$\mathbb{P}(W_1 \mid S)=\frac{\mathbb{P}(W_1 \cap S)}{\mathbb{P}(S)}=\frac{\mathbb{P}(W_1 \cap W_4)}{\mathbb{P}((W_1 \cup W_2 \cup W_3)\cap W_4)}=\frac{1}{(p_1 + p_2 + p_3)p_4}.$$

Is this correct or have I missed anything? Thank you very much.

$\endgroup$
2
  • $\begingroup$ Are components 1 through 4 working independent events? $\endgroup$
    – paw88789
    Apr 22, 2018 at 13:11
  • $\begingroup$ Yes! I edited the original post. $\endgroup$
    – Olegas Rudgalvis
    Apr 22, 2018 at 14:20

1 Answer 1

Reset to default
0
$\begingroup$

The probability of $P(W_1\cup W_2\cup W_3)$ is not equal to the sum of the probabilities ($p_1+p_2+p_3$), as probabilities of overlaps would be included more than once.

One method to find the probability of a union is to use the Principle of Inclusion-Exclusion. (See for instance https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle).

$\endgroup$
2
  • $\begingroup$ The probability of any component working or failing is independent of any other, and if I understand it correctly, there would be no overlap. Please correct me if I'm wrong, just trying to understand this. $\endgroup$
    – Olegas Rudgalvis
    Apr 22, 2018 at 15:22
  • $\begingroup$ No overlap would mean the events are mutually exclusive. Independence is very different. $\endgroup$
    – paw88789
    Apr 22, 2018 at 17:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.