Four of a kind given at least three of a kind

A five-card hand is dealt from a standard $52$-card deck. You are informed that at least three of the cards share a rank. What is the conditional probability that the hand holds all four cards of some rank?

$1/95$. Four-of-a-kind hands: $13\cdot48=624$. Hands with exactly three of a rank: $13\binom43\binom{48}{2}=58656$. The answer is $\tfrac{624}{624+58656}=\tfrac{1}{95}$.