
Suppose that there are $n$ events possible. Let $A$ be a set of $n - k$ arbitrary events and $B$ the set of other $k$ events. Then $\rm A \cup B$ denotes all the events possible.
$P({\rm A \cup B})$ denotes the probability of an event happening from either of set $\rm A$, or $\rm{B}$ (which would be $1$ in this case since all events possible are contained within $\rm A \cup B$).
Example. Suppose we are throwing a fair dice. Let $\rm O$ denote the set of events where the number we get is odd. Let $\rm E$ denote the set of events where the number we get is even. It is obvious that,$\begin{aligned} \rm O &= \{1,3,5\} \\ \rm E &= \{2,4,6\} \\\rm O \cup E & = \{1,2,3,4,5,6\} \end{aligned}$
Property. If $\rm A \cap B = \emptyset$, then $\rm P(A \cup B) = 1$ (if $A$ and $B$ are the only sets).
Pop Quiz.
- What do you infer from $\rm P(A \cap B)$?
- Can you find other properties for different cases?