SIG (Susquehanna)–styleTrader InternQuant Trader InternQuestion #92
A deck contains ten cards: two 10s, two Js, two Qs, two Ks, and two As. You are dealt five cards without replacement from the deck. Compute the expected number of pairs.
Free worked solution · Quant Blueprint
10/9 complete rank pairs on average.
- Define one indicator for each of the five ranks, equal to one if both cards of that rank are in the hand.
- For one rank, the chance both cards appear is (5/10)(4/9) = 2/9. Equivalently it is choose(8,3)/choose(10,5).
- Add the expectations for five ranks: E(pairs) = 5 × 2/9 = 10/9. No independence between ranks is needed.
Common mistake
The question asks for the number of pairs, not the probability of having at least one pair. A hand can contain two pairs.
Follow-up
The only no-pair hands take one card of each rank: 2⁵ possibilities out of choose(10,5) = 252. Thus P(at least one pair) = 1 − 32/252 = 55/63.
Original educational explanation for this practice question. Reviewed . Firm-style practice; the label describes the collection’s preparation context.
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