Last red marble leaving blue and green behind

An urn holds $r$ red, $b$ blue and $g$ green marbles, which Josephine removes one by one at random without replacement. Find the probability that, immediately after the final red marble is removed, the urn still contains at least one blue and at least one green marble. Then evaluate it for $r=10$, $b=20$, $g=30$.

$\dfrac{g}{r+b+g}\cdot\dfrac{b}{r+b}+\dfrac{b}{r+b+g}\cdot\dfrac{g}{r+g}$, which is $7/12$ here. Reds must run out first. Split on which colour supplies the very last marble: if green, you also need the last of reds-and-blues to be blue; if blue, the last of reds-and-greens must be green. With the numbers: $\tfrac12\cdot\tfrac23+\tfrac13\cdot\tfrac34$.