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I have not taken a subject in set theory, only in statistics. Maybe you guys can help me, I want to describe one of the events with the other, and I am wondering if one can do it.

Lets say that we have 6 machines, of these 0, 1,2,3,4,5 or 6 of them can be in use. That is, we distingiuish how many are in use, not which particular one we are using.

Now lets say that you want to find the probability that "not 2 or 3 or 4 machines are in use".

If we say that the event A is {2 machines, or 3 machines or 4 machines in use}

then we want to find P(not A) = 1-P(A), this is easy.

Now is the tricky part, look at the event:

"2,3 or 4 machines are not in use". First I thought that this was the same as the first one, but it is actually the same as the event A, because if 2 is not in use, then 4 is in use, and if 3 is not in use then 3 is in use, and if 4 is not in use then 2 is in use.

So we have that "2,3 or 4 in use" = "2,3,4 not in use" and this does not equal " not 2,3,4 in use"

My question is if this can be shown with something deeper, than just going over all the different possibilities? For instance, could we say that the event "2,3 or 4 machines are not in use" is "not not A" and hence it becomes A? Or is it another way to describe "2,3 or 4 machines are not in use" in terms of A, and then reduce this expression so you get to A?

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# Description of events

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