Exercise: A forest contains $100$ deer. $20$ of them have a red tag and $80$ of them are untagged. A researcher samples $30$ random deer without replacement. Let $X$ be the number of tagged deer in the sample. From the sample of $30$ deer, she will keep picking deer with replacement until she picks the fourth tagged deer. Let $Y$ be the number of selections she makes until she gets her fourth tagged deer. Find the joint pmf of $X$ and $Y$.
I tried to think that: $$P_{xy}(X=x,Y=y)=P(X=x)P(Y=y\ |\ X=x)=\frac{{20\choose x}{80 \choose 30-x}}{100 \choose 30}.{{y-1}\choose 4-1}(0.2)^4(0.8)^{y-4}$$ $x\in \{4,5,6,....,30\}$ and $y\in \{4,5,6,7,.....30\}$
Can that be a viable conclusion? If not is there a better way to tackle this kind of problems ?