Chance the top two cards share a rank

Question

A standard 52-card deck is shuffled thoroughly. What is the probability that the first two cards on top have the same rank (for example, two sevens)?

Short answer

$1/17$ . Whatever the top card is, exactly $3$ of the remaining $51$ cards match its rank, so the chance the second card pairs it is $\tfrac{3}{51}=\tfrac{1}{17}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Matrix Algebra: Operations, Rules, and the Transpose
Kernel, Image, and the Rank-Nullity Theorem

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