Two cards, same rank?

Question

You deal the top two cards from a well-shuffled standard 52-card deck. What is the probability the two cards have the same rank (for example, two Kings or two Sevens)?

Accepted Answer

The fastest route is conditional. The first card can be anything. For a match, the second card must share its rank. After the first card is dealt, 51 cards remain, and exactly 3 of them share the first card's rank: P(	ext{pair}) = \frac{3}{51} = \frac{1}{17} \approx 0.0588. A counting check: there are \binom{52}{2} = 1326 two-card hands. Same-rank hands are 13 	imes \binom{4}{2} = 13	imes 6 = 78. Then \frac{78}{1326} = \frac{1}{17}. Takeaway. "Do the two match" often collapses to a single conditional step: fix the first outcome, then count the survivors that match it. Here 3 of the remaini…

Two cards, same rank?

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