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Expected Number of Same-Rank Pairs in a Hand Dealt Without Replacement

SIG (Susquehanna)Quantitative Trader · Online Assessment
MediumQuantitative TraderOnline AssessmentStatistics & Math

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An expectation question about dealing K cards without replacement from a deck of M ranks with N cards each, then counting the same-rank pairs in the hand. It tests linearity of expectation, probabilities when drawing without replacement, and how the definition of a pair changes the answer.

A deck contains $M$ ranks with $N$ cards of each rank, so it holds $MN$ cards in total. You are dealt $K$ cards from the shuffled deck, without replacement. What is the expected number of pairs in your hand?

Count a pair as any two cards in the hand that share a rank, so three cards of one rank contain three pairs. Give an expression in $M$, $N$ and $K$, then evaluate it for the values you are given; the specific numbers are generated randomly. Also say how the answer changes if a pair instead means a rank that appears exactly twice in the hand.

Constraints and Clarifications

  • $M$, $N$ and $K$ are positive integers, and $K$ is at most $MN$.
  • Every set of $K$ cards is equally likely to be dealt.

Clarifying Questions

  • Does three of a kind count as three pairs, as one pair, or as no pair?
  • Should four cards of one rank count as six pairs, as two disjoint pairs, or as no pair?
  • Should the answer be exact or rounded to two decimal places?

Hints

Hint 1

Skip the full distribution: You do not need the probability of every possible hand. Look for small events whose indicators add up to the number of pairs.

Follow-up questions

  • What is the expected number of ranks that appear exactly twice in the hand?
  • What is the probability that the hand contains no pair at all?
  • How does the answer change if the cards are dealt with replacement?