A copper cable has an average of $0.002$ flaws per metre. A fibre optic cable has an average of $0.003$ flaws per metre. $100$ metres of copper cable and $100$ metres of fibre optic cable are installed in the building. What is the probability that there are no flaws length of copper and fibre optic cable?
What I've done is using the Poisson distribution to calculate the probability of there being no faults in either wire. Now what I want to do is multiply these probabilities, I feel that it's right, but I feel like 1) multiplying the probabilities that there are no faults and 2) multiplying the probabilities there are faults and taking away one. But this doesn't give the same answer? Are any of them right? If so, why is one right and not the other?
If $C$ denotes the event that there are no flaws in copper cable and $F$ denotes the event that there are no flaws in fibre optic cable then $C\cap F$ is the event that there are no flaws in both cables, and by independence: $$P(C\cap F)=P(C)P(F)$$
Next to that by independence: $$1-P(C^{\complement})P(F^{\complement})$$ is the probability of: $$(C^{\complement}\cap F^{\complement})^{\complement}=C\cup F$$which is the event that in at least one of the cables there are no flaws.