Purple cherries left when the last red is eaten

A bowl holds $8$ purple and $5$ red cherries. Emilia removes them one at a time in random order until the bowl is empty. What is the probability that exactly $2$ purple cherries remain in the bowl at the moment she takes out the final red cherry?

$\tfrac{70}{429}$. This happens exactly when the last three cherries in the random order are red, purple, purple. Reading the order backwards, that has probability $\tfrac{8}{13}\cdot\tfrac{7}{12}\cdot\tfrac{5}{11}=\tfrac{70}{429}$.