Apples running out before oranges and peaches

A basket contains 10 apples, 20 oranges and 30 peaches. Fruit is removed one piece at a time, uniformly at random. At the moment the last apple is removed, what is the probability that at least one orange and at least one peach are still in the basket?

$7/12$. We need apples to be exhausted first. If the very last fruit is a peach (prob. $\tfrac12$), we also need the last of the apples-and-oranges to be an orange ($\tfrac23$); if the last fruit is an orange (prob. $\tfrac13$), we need the last of apples-and-peaches to be a peach ($\tfrac34$). Total $\tfrac13+\tfrac14=\tfrac{7}{12}$.