Q.The probability that a new railway bridge will get an award for its design is 0.48, the probability that it will get an award for the efficient use of materials is 0.36, and that it will get both awards is 0.2. What is the probability that

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 $A,B$:

$P(A\cup B)=P(A)+P(B)-P(A\cap B).$ …