Let $A$ and $B$ be events in the probability space $(\Omega, \mathcal{F}, \mathcal{P})$. How would I write that
events $A$ and $B$ both occur
and
event $A$ occurs with probability $0.5$ and the event $B$ does not occur
and
neither of the events occurs?
An event $A$ (a subset of $\Omega$ that is also a member of $\mathcal F$) is said to occur if and only if the outcome of the experiment is an element that belongs to $A$. The outcome $\omega$ of the experiment is always one element of the sample space $\Omega$, but the occurrence of this outcome $\omega$ leads to many events occurring because $\omega$ is a member of numerous subsets of $\Omega$ that are in $\mathcal F$. In particular, regardless of what $\omega$ is, one and only one of the events $A$ and $A^c$ must occur because $\omega$ necessarily belongs to exactly one of $A$ and $A^c$; it cannot belong to both, and it cannot be excluded from both. If $\mathcal F$ is a finite collection of subsets (there must be $2^n$ of them for some integer $n$), then half the events occur and half don't, regardless of what $\omega$ is. In some sense, this is also true when $\mathcal F$ has infinitely many subsets: there is a one-to-one correspondence $A \leftrightarrow A^c$ between those events that occur and those that don't. $\Omega$ always occurs and so is called the sure or certain event, while its complement $\emptyset$ never occurs ans so is called the impossible event (also the null event by some, but there are many, including myself, who deprecate this usage).
With that as prologue, if you are told that events $A$ and $B$ both have occurred, then the outcome $\omega$ must be a member of both $A$ and $B$, right? What is the notation for the set of elements that are members of both $A$ and $B$?
