Red candies running out before blue and green

A jar contains $10$ red, $20$ blue and $30$ green candies, which are taken out one by one in uniformly random order. What is the probability that when the final red candy is removed, the jar still holds at least one blue and at least one green candy?

$7/12$. We need red to be the first colour to run out. Case 1: the last candy is green ($\tfrac{30}{60}$) and, among red and blue, the last is blue ($\tfrac{20}{30}$). Case 2: the last candy is blue ($\tfrac{20}{60}$) and, among red and green, the last is green ($\tfrac{30}{40}$). Total $\tfrac13+\tfrac14=\tfrac7{12}$.