Say we have $r$ red balls, $b$ blue balls, and $g$ green balls in a bag. I want to know the probability of red balls being depleted, then blue balls being depleted, and finally green balls being depleted if we continually draw a random ball from the bag until we run out of balls. I know that the probability of red being depleted before blue is: $b/(r+b)$, and likewise probability of blue being depleted before green is $g/(b+g)$.
Are these two events (red being depleted before blue and blue being depleted before green) independent (in which case I just have to multiply the two probability together)? If they are not how could I go about calculating the probability of blue being depleted before green given red depleted before blue?
You have determined that in a bag with $x$ balls of color X and $y$ balls of color Y, the probability of depleting the balls of color X first is $$\frac{y}{x+y}$$ Notice now that in a bag of red, blue, and green balls, depleting the balls of each color in that order is the same as depleting the red balls before the blue balls, and depleting the non-green balls before the green balls. While the two events that you described are not independent, these two events are independent. Thus, since the probability of depleting red before blue is $$\frac{b}{r+b}$$ and the probability of depleting red and blue before green is $$\frac{g}{r+b+g}$$ and so the probability of depleting red, blue, and then green in that order should be $$\frac{b}{r+b}\cdot \frac{g}{r+b+g}$$