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I pick 5 cards uniformly at random from a deck of 52 cards (13 different values for 4 different suits). I tell you that the first card I drew was a 4 of hearts. What is the probability that my hand contains 4 of a kind? (Four cards of the same value in different suits.)

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    Any thoughts? Hint: handle the cases "four $4's$" and "four non-$4's$" separately. – 2017-02-26
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    Alternate hint: does it matter which card you draw first? – 2017-02-26

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There are two cases:

If you obtained $3$ other $4$'s in your hand, then the probability that this could have happened is $\frac{\binom{4}{3}\cdot48}{\binom{51}{4}}$, since there are $\binom{4}{3}$ ways of choosing the $3$ cards and there are $48$ other cards in the deck.

If you obtained a $4$-of-a-kind in another rank, the probability is $\frac{12}{\binom{51}{4}}$, since there are $12$ other ranks to choose from.

Therefore, the answer is $\frac{\binom{4}{3}\cdot48+12}{\binom{51}{4}}=\frac{1}{1225}$.

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    Rather than just telling the whole answer, you should leave it at telling how to get the answer. Let the student do some work on their own. – 2017-02-26
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    i think this not true, why to obtains 3 other 4's in the hand is 4 choose 3? – 2017-02-28