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.
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).