Prove the Probability of Two Events

Solution 1:

For any sets $A$ and $B$, we have the disjoint union

$$A \cup B = (A- B) \cup (A \cap B) \cup (B-A). $$

Then, by the axiom,

$$ P (A \cup B ) = P(A- B) + P(A \cap B) + P(B - A). $$

Since $ P (A-B)= P(A) - P(A \cap B)$ and $ P (B-A)= P(B)- P(A \cap B) $, the result follows.

Here's how I'd do it:

Note that $$ (*)\quad P(A\bigcup B) = P(A \bigcup A^{c}B) = P(A)+P(A^{c}B)$$

Then since $B = AB \bigcup A^{c}B$ we get that

$$P(B) = P(AB) + P(A^{c}B)$$

or equivalently $$P(A^{c}B) = P(B) - P(AB)$$ Plugging this into $*$ gives the desired result and thus completing the proof. Hope that helps!