Consider a sequence of independent Bernoulli trials, each of which is a success with probability $p$. Let $X$ be the number of failures preceding the first success, and let $Y$ be the number of failures between the first two successes. Find the joint mass function of $X$ and $Y$.
I struggle in differing the number of failures $X+Y$ and the number of trials $W$, because $W=X+Y+2$ has negative binomial distribution but I don't know how to relate it properly with $P(X=x,Y=y) = p(x,y)$.
I came up with the answer $p(x,y)=P(X=x,Y=y)=P($x+y+2 trials until 2 successes$)=\binom{x+y+2}{1}p^2(1-p)^{x+y}=(x+y+1)p^2(1-p)^{x+y}$
But the correct answer is $p(x,y)=p^2(1-p)^{x+y}$.
I think there is a way to count the possibilities I'm adding beyond the required but I can't see how to do it properly because of the relation of the two variables and the variable that has the distribution used.