## Proof related to Axioms of Probability

These problems and Proofs are adapted from the textbook: Probability and Random Process by Scott Miller 2ed Problem-1: Proof that for events A and B the following holds: $$Pr\left(A\cup B\right) = Pr\left(A\right)+Pr\left(B\right)-Pr\left(A\cap B\right)$$ Solution: First, note that the sets $$A\cup B$$ and $$B$$ can be rewritten as $$A\cup B=\{A\cup B\}\cap\{\overline{B}\cup B\}=\{A\cap\overline{B}\}\cup B$$ $$B=B\cap\{A\cup\overline{A}\}=\{A\cap B\}\cup\{\overline{A}\cap B\}.$$ Hence, $$A\cup B$$ can be expressed as the union of three mutually exclusive sets. $$A\cup B=\{\overline{B}\cap A\}\cup\{A\cap B\}\cup\{\overline{A}\cap B\}$$ Next, rewrite $$\overline{B}\cap A$$ as $$\overline{B} \cap A= \{\overline{B}\cap A\}\cup\{\overline{A}\cap A\}= A\cap\{\overline{A}\cup\overline{B}\} = A\cap\{\overline{A\cap B}\}$$ Likewise, $$\overline{A}\cap B=B\cap\{\overline{A\cap B}\}$$. Therefore $$A\cup B$$ can be rewritten as the following...