What does it mean to be event {N=2} in probability?

Question

I am studying but I don't get this.

The event {N = 2} is the union of the following events {N = 2}= [A∩B∩Cc]∪[A∩C∩Bc]∪[B∩C∩Ac]

I just don't know what it mean to be event {N = 2}.

Also, then, what does it mean if it were event {N = 1} or {N = 3}, and so on? I probably need detailed explanation by many cases.

Thank you very much.

Answer 1

So, after taking consideration your comment, {N = 2} means that only two of the events occurred out of your three events A, B, and C. That means either A and B occurred and C didn't, A and C occurred and B didn't, or B and C occurred and A didn't. That is what that expression with unions and intersections signifies.

To understand why they are using unions and intersections, you kind of need a good definition of an event, which is defined as a set of outcomes$^*$. Let's say that my experiment is tossing a die, so my outcomes are 1,2,3,4,5,6. The event that the die is even is the set of outcomes {2,4,6}, the event that the die is bigger than or equal to 3 is the set of outcomes {3,4,5,6} and so on.

Note that unions and intersections of events are still events (since if you add or remove outcomes from an event, you are -by the current definition- still an event), and they correspond to "or" and "and". (So, if I asked you what event has the die is even and bigger than or equal to 5, you would look at the outcomes for both, and see where they intersect, so it would be {2,4,6} $\cap$ {3,4,5,6} = {4,6}.)

So, try to do N=3; what does it mean that A, B, and C all happened? That means that some outcome that is in A, B and C must have happened; can you write that in terms of unions and intersections?

$^*:$ If you study maths, you will see that not every set of outcomes would be considered an event, but only those that we can "measure", but that is a huge technicality for you right now.

Answer 2

Thanks for your comment for the definition of $N$.

$N$ here means "how many events have been happened".

For example, if A happens but B and C do not happen, then $N=1$.

So if we want to know the case when $N=2$, we better have:

1) A and B happen, but C does not
2) A and C happen, but B does not
3) B and C happen, but A does not

Then if you know that $X^c$ means "$X$ does not happen", $\cap$ means "and" and $\cup$ means "or", you can understand what is meant by that mysterious notation.

(P.S. I have listed so that it is exactly in the sequence in your question so that possibly it is easier for you to follow)