Q.The probability of an event $A$ occurring is 0.5 and $B$ occurring is 0.3. If $A$ and $B$ are mutually exclusive events, then find the probability of

Working directly from the three axioms of probability, four theorems give the tools used to combine probabilities of related events.

Theorem 12.3 (impossible event). $P(\varnothing)=0$ -- since $S=S\cup\varnothing$ with $S,\varnothing$ mutually exclusive, axiom $[P_2]$ gives $P(S)=P(S)+P(\varnothing)$, so $P(\varnothing)=0$.

Theorem 12.4 (complement rule). $P(\bar A)=1-P(A)$ -- since $A\cup\bar A=S$ with $A,\bar A$ mutually exclusive, $P(A)+P(\bar A)=P(S)=1$.

Theorem 12.5 (only-$A$ rule). $P(A\cap\bar B)=P(A)-P(A\cap B)$ -- the part of $A$ that misses $B$ has probability 'all of $A$' minus 'the overlap with $B$', since $(A\cap\bar B)\cup(A\cap B)=A$ with the two pieces mutually exclusive.

Theorem 12.6 -- the Addition Theorem. For any two events $A,B$:

$P(A\cup B)=P(A)+P(B)-P(A\cap B).$ …