One of each face before the two of hearts

HardProbability Combinatorics~9m

A standard $52$ -card deck is shuffled and dealt one card at a time until the $2\heartsuit$ appears. What is the probability that exactly one King, exactly one Queen, and exactly one Jack appear before the $2\heartsuit$ ?

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Only the $2\heartsuit$ and the $12$ face cards matter. First: probability exactly $3$ faces precede the $2\heartsuit$ is $1/13$ (its position is uniform among $13$ ).
Then those $3$ are a random $3$ -subset of the $12$ faces; chance of one each is $\binom41^3/\binom{12}{3}$ .

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0.0224 (± 0.001)

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One of each face before the two of hearts

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