Carousel Box I

Question

A box has $1$ red and $1$ blue ball. $3$ people named $1,2,$ and $3$, draw from the box one-at-a-time with replacement in that order. A round will be considered as the process of people $1,2,$ and $3$ drawing consecutively. Let $X$ be the number of the person who selects the first red ball and $Y$ be the round in which the first red ball is drawn. For each valid value of $x$ and $y$, compute $\prob{X = x, Y = y}$. Evaluate this when $x = 1$ and $y = 2$.