I'm struggle in undestanding condinal probability formula $ P(E|F)= \frac {P(EF)}{P(F)}$
I thought that the formula tell us to restrict the sample space of events $E$ and make F to be a new sample space for E, then we calculate the $P(EF)$ on $F$, since $P(E|F)= \frac {|EF|}{|F|} = \frac {\frac{|EF|}{|S|}} {\frac {|F|}{|S|} } = \frac {P(EF)}{P(F)}$, which tell us that E and F both lie on S. But in this problem
"Suppose that we have 3 cards that are identical in form, except that both sides of the first card are colored red, both sides of the second card are colored black, and one side of the third card is colored red and the other side black. The 3 cards are mixed up in a hat, and 1 card is randomly selected and put down on the ground. If the upper side of the chosen card is colored red, what is the probability that the other side is colored black?"
Let RR, BB, and RB denote, respectively, the events that the chosen card is all red, all black, or the red–black card. Also, let R be the event that the upturned side of the chosen card is red. Then, the desired probability is obtained by $P(RB|R)= \frac {P( RB \cap R) } {P(R)}$
$P(R)= 1/2$ if we choose sample space as 6 sides of 3 cards $S$={${R_1,R_2,B_1,B_2,R_3,B_3}$}. But i cant calculute $P(RB \cap R)$ using the same sample space S. Is there exist a sample space that i can calculate both events on it?
I have one more misunderstanding is could F be in another sample space? Suppose E denote:" The rolled dice is < 4" for 6-side dice with $S_1$ ={$1,..,6$} and F denote :" The rolled dice is < 8 " with 12-side dice but I mark the number on it start at 2 $S_2$={$2,...,13$}. Can I choose a new sample space $S=S_1 \times S_2$ to calculate?