Q.A town has 2 fire engines operating independently. The probability that a fire engine is available when needed is 0.96.

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 $A,B$:

$P(A\cup B)=P(A)+P(B)-P(A\cap B).$ …