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CAF123
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Homework Statement
A parallel system functions whenever at least one of its components works. Consider a parallel system of ##n## components, and suppose that each component works independently with prob 1/2. FInd the conditional prob that component 1 works given that the system is functioning.
The Attempt at a Solution
Let E be the event that compt 1 works and F the event that the system functions.
P(E|F) = P(EF)/P(F) = P(F|E)P(E)/P(F). I have the numerator correct, I just have a subtle question about the denominator.
When I see the words 'at least', I always jump to the conclusion of doing 1-P(less than 'at least'). Is this a normal practice?
So I get the correct answer for the denominator using this way. However, I also considered finding the denominator by conditioning on the component that works. So I said;
Let ##G_i## be the event that compt i works. Then $$P(F) = P(\cup_{i=1}^{n} F|G_i) = P(F|G_1)P(G_1) + P(F|G_2)P(G_2) +...+ P(F|G_n)P(G_n)$$. Correct me if I am wrong, but I believe this method fails because this assumes that the system functions when only one component works? In general, a union is where we want only one event to happen out of,say, n possibilities? Thanks!
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