What does P(A U B) mean, in terms of real values? I can't find a proper summary or reference of how to translate formulas in probability notation to arithmetic notation (i.e. when using real values).
For example, if $P(A) = .7$ and $P(B)=.35$, what does $P(A \cup B)$ translate to? 
What does $P(A \cap B)$ translate to?
etc...
 A: That depends on how the sets $A$ and $B$ intersect. 
For example, suppose your probability space is interval $[0,1]$ and probability density is uniform. 
If $A=[0,0.7]$ and $B=[0,0.35]$ then $A\cup B=[0,0.7]$ and $P(A\cup B)=0.7$. 
On the other hand, if $A=[0,0.7]$ and $B=[0.65,1]$ then $A\cup B=[0,1]$ and $P(A\cup B)=1$.
A: $P(A\cup B)$ is the probability that the event is in $A$ or $B$. For example, if your space of events is $\{1,2,3,4,5,6\}$ (like throwing a dice), define $A=\{1,2\}$ and $B=\{6\}$. In that case, $P(A\cup B)$ is the probability that the dice gives you $1,2$ or $6$. Therefore $P(A\cup B) = \frac{3}{6} = \frac{1}{2}=0.5=50\%$.
For intersection or others, the idea is the same. In general $P(X)$ is the probability of an event in $X$ happens after the experiment is made, whatever it is.
A: $P(A \cup B)=P(A) + P(B) - P(A \cap B) $
When $A$ and $B$ are incompatible events; that is $A \cap B= \emptyset$:
$P(A \cup B)=P(A) + P(B) $
$\\$
$P(A \cap B)$ is the probability of both $A$ and $B$ to happen.
$P(A \cap B)=P(A) \cdot P(B)$ 
