I realize this is a rather trivial question when it comes to some that are asked on this exchange, but I was hoping for an intuitive (AND mathematical - I can't make sense of it if I don't understand the math) explanation of this problem:
This is actually an example problem in my book and I don't really understand the answer in non-mathematical or mathematical terms.
Roll two fair four-sided dice. Let $X_1$ and $X_2$ denote the number of dots that appear on die 1 and 2, respectively. Let A be the event $X_1 \geq 2$. What is P[A]? Let B denote the event $X_2 > X_1$. What is P[B]? What is P[A|B]?
So in my mind, the answer to what is P[A] = 3/4 - BECAUSE, die 1 can either be 2,3, or 4. BUT, according to their answer it is 12/16 = 3/4. Now, this is counter-intuitive to me because die 2 does not have anything to do with P[A] in this question..? (Note: They drew a diagram of 4x4 'grid' and said $X_2$ could be any of it's combinations.) This is somewhat bothersome to me, but the most bothersome is the second and third part of the question:
My response to P[B]?
If $X_1$ = 1, $X_2$ = 2,3,4 => 3/4
If $X_1$ = 2, $X_2$ = 3,4 => 2/4
If $X_1$ = 3, $X_2$ = 4 = 1/4
We then add 3/4 + 2/4 + 1/4 = 6/4 - which is of course wrong.
What exactly am I missing here? I understand that there are 16 combinations, but it doesn't seem to me that they matter at this point? The book uses a graph to explain this, but it is more or less equivalent to: 3/16 + 2/16 + 1/16 = 6/16 = P[B]. Why is it $a$/16 since we are only looking at die 2 with respect to die 1. That is, if a 1 is rolled on die 1, die 2 can either be a 2,3, or 4 (out of it's four possibilities)- which is why I have my error from above..
Then, we have P[A|B]. So, P[A|B] = $P[A\cap B]\div P[B]$. Let us assume that this problem was huge and I could not say (2,3),(2,4),(3,4) = 3/16 for P[AB] how then, would one easily determine P[AB]?
You are my hero if you are able to guide me in the probabilistic way of thinking on this!
Thanks in advance!