Hi all, i'm preparing for an exam and I'm having difficulty interpreting this problem, nevermind solving it. What I'd like is for someone to sort of "translate" into words what question B) and C) are asking. Can you explain what is required?

I'm in a weird situation... it's a math stats exam. I've never been strong in "probability/set theory" and I really could use the extra help on this chapter (our first chapter of the course lol!). Everything else I seem to get, but its these questions take a bit more time for me to understand. Maybe someone can literally spell it out to help me understand what the question is asking, maybe with a real life example? Please see attached.

I can't really think of a real life example...but then I try not to worry about real life when I'm in math. But I'll try to be helpful to doing this problem.

Have you done proofs before? "if and only if" proofs? If so, then you know that one way to do them is to suppose the first half is true, then show (ie prove) that the second half follows. Then conversely you suppose the second half is true and then show that the first half follows.

Here's how I'd do part b)
Suppose that omega is in the intersection of B1,B2,... [we wish to show that omega is in an infinite number of the events A1,A2,...]
Just suppose that omega is in only a finite number of the events Ai. Then there is some integer N such that omega is NOT in A_N+1, A_N+2,... This would mean that omega is NOT in B_N+1, nor in B_N+2, nor in B_N+3, etc.
[now you have a contradiction, because of what you first supposed]
But this is impossible since omega is in the intersection of B1,B2,... Therefore omega must be in an infinite number of the events Ai.

On the other hand, now suppose that omega is in an infinite number of the events Ai. Then omega must be in B_N for any integer N. [look at the way B_n is defined]. Therefore omega is in the intersection of B1,B2,...

This was exactly what I was looking for! I have the solutions to these problems by the prof (back from earlier in the term) but I really wanted to do them without looking at them. That is, try and understand them as opposed to memorizing them. What you have written is very helpful. I am going to try part C by myself.

Again, I'm in a bit of a weird situation. I'm in a biostatistics program and as a part of our requirements we must have one full year of mathematical statistics. I come from a background in Psychology, so I've missed a lot of mathematical fundamentals that would have otherwise made this course more easier (proofs, linear algebra, real analysis and even comp sci to some extent).

Thank you so much again, I really appreciate this.