Two Kings

You are dealt exactly two cards from a well-shuffled standard 52-card deck, which contains exactly four kings. What is the probability that both of your cards are kings?

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#Two dependent draws

The first card is a king with probability $\tfrac{4}{52}$ . Given that it was a king, three kings remain among the $51$ cards left, so the second card is a king with probability $\tfrac{3}{51}$ . The two must happen together, and dependent events chain by multiplication,

\PP(\text{both kings}) = \frac{4}{52} \cdot \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}. \tag{1}

A king first costs four chances in fifty-two, and a second king then three in the fifty-one cards that remain. Multiplying the two dependent draws gives one chance in two hundred twenty-one.

#A counting check

Choosing $2$ of the $4$ kings out of all $2$ -card hands gives the same value,

\PP(\text{both kings}) = \frac{\binom{4}{2}}{\binom{52}{2}} = \frac{6}{1326} = \frac{1}{221}. \tag{2}

#Result

\PP(\text{both kings}) = \frac{1}{221} \approx 0.452\%, \tag{3}

about one hand in $221$ .