Show that $p+q-1 \le r \le min(p,q)$

Question

I have the following problem.

Let X and Y be Bernoulli random variables with parameters p and q, respectively. The random variable Z = XY also takes only values 0 and 1 and hence is a Bernoulli random variable with some parameter r. Show that $$p+q-1 \le r \le min(p,q) $$

Hint: Use the fact that Z = 0 when X = 0 or Y = 0.

Any hint, suggestion or idea is welcomed.

Thank you

Answer 1

In general, if an event $A$ is contained in an other event $B$, then $\mathbb{P}(A) \leq \mathbb{P}(B)$.

Note that if $XY=1$, then $X=1$. Hence $\{XY=1\} \subset \{X=1\}$. What does this tell you about $p$, $q$ and $r$?

You also know that $\{XY=0\} \subset \{X=0\} \cup \{Y=0\}$ (why?). What can you deduce from this?