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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.

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  • $\begingroup$ Are components 1 through 4 working independent events? $\endgroup$ – paw88789 Apr 22 '18 at 13:11
  • $\begingroup$ Yes! I edited the original post. $\endgroup$ – Olegas Rudgalvis Apr 22 '18 at 14:20
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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).

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  • $\begingroup$ 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 '18 at 15:22
  • $\begingroup$ No overlap would mean the events are mutually exclusive. Independence is very different. $\endgroup$ – paw88789 Apr 22 '18 at 17:55

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