I am just looking for some guidance whether I am approaching the questions correctly. New to probability!
Prove that if $A$ and $B$ are events, then $$\Pr[A \cap B] \geq \Pr[A]+\Pr[B]-1$$
I rewrote the statement in the following way :
$$1 \geq \Pr[A] + \Pr[B] - \Pr[A \cap B]$$
And based on some probability properties, I rewrote out the statement again:
$$1 \geq \Pr[A \cup B]$$
Is this enough to prove the statement?
So you use $$P(A\cup B ) = P(A)+P(B)-P(A\cap B)$$ and that $P(A\cup B)\leq 1$
So you got $$1\geq P(A)+P(B)-P(A\cap B)$$ and now swap $1$ and $P(A\cap B)$ and you got what you want.
That is correct.