A parallel system functions whenever at least one of its components works. Consider a parallel system of three components, and suppose that each component works independently with probability $0.5$.

Find the conditional probability that component 1 works given that the system is functioning.

lets say $A_i =$ the event that the $i$th component works. If they're not mutually exclusive, what would $A_1 \cap A_2$ be??

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    $\begingroup$ Your question title doesn't seem to match your question. $A_1 \cap A_2$ is the event that the $1$st an $2$nd components both work. $\endgroup$ – Clive Newstead Sep 20 '12 at 16:08
  • $\begingroup$ for two events to be mutually exclusive $A_1$ $\cap $ $A_2$ must be nothing. However, I've been told in this question, those two events are not mutually exclusive. I just need some evidence to show that they're not mutually exclusive... $\endgroup$ – user133466 Sep 20 '12 at 16:10
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    $\begingroup$ Mutual exclusivity would mean that if $A_1$ works then $A_2$ cannot; is there anything in the question that implies that this is so? [In fact, there's something in the statement of the question that implies that it isn't so!] $\endgroup$ – Clive Newstead Sep 20 '12 at 16:12
  • $\begingroup$ what statement implies that it isn't so? $\endgroup$ – user133466 Sep 20 '12 at 16:26
  • $\begingroup$ The events are independent (and have nonzero probabilities). $\endgroup$ – Clive Newstead Sep 20 '12 at 16:27

If $A_1$, $A_2$ are independent, then $$\Pr(A_1\cap A_2) = \Pr(A_a)\cdot\Pr(A_2) = (0.5)(0.5) = 0.25 \ne 0.$$ If $A_1$ and $A_2$ were mutually exclusive, then that probability would be $0$.

The question in your title asks why they are not mutually exclusive. The above should answer that.

In the body of your question you ask what $A_1\cap A_2$ is. It's just the event that the first two components both work.

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  • $\begingroup$ this is a good answer Thanks!!! $\endgroup$ – user133466 Sep 20 '12 at 16:34

For the conditional probability, you can see that there are 8 equally likely outcomes possible for the functionality of the components, only one of which results in the system not working.

Of the seven working systems, three have component #1 not working and four have component #1 working.

So, the condition probability that component #1 is working, given that the whole system is working is $4/7$.

(This whole setup is just like a three coin flip experiment, where you are asking "If a friend flipped a coin three times and told you at least one head appeared, what is the probability a heads appeared on your friend's first flip?"

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