Probability of Specific Poker Hand Combinations

This question asks about computing the likelihood of several classic five-card poker hands from a standard deck, focusing on hands defined by repeated card values: four-of-a-kind, full house, and two pairs. The candidate must translate the informal descriptions of these hands into precise counting problems: how many distinct hands match each pattern, and how that compares to the total number of five-card hands. The scenario reflects standard probability setups seen in interviews for roles that expect comfort with discrete probability and basic combinatorics, and it is particularly common in quantitative finance and tech interviews that probe foundational reasoning under simple card models.

To answer correctly, a candidate needs fluency with combinations, counting partitions of card ranks, and systematically handling suit choices once value patterns are fixed. It leans on understanding equivalence classes of hands (where order of cards is irrelevant) and careful separation of "choose ranks" versus "assign suits" steps without double-counting. Interviewers watch for a clear, structured counting argument, correct decomposition of the problem into independent choices, and the ability to check results against sanity expectations, such as relative rarity of different hand types.

What it tests

The core structure in enumerating poker hands is combinatorial partitioning: each hand type is defined by a partition of the cards into equivalence classes based on value (rank), and then, within those classes, by the selection of suits. The process is hierarchical: first, the hand is classified by the distinct card values it must contain and their multiplicities (e.g., one value appears four times, another appears once), and only then are the suits for those values chosen. This partitioning is governed by the principle of counting distinguishable arrangements: for each possible combination of values, the number of suit arrangements is determined by the multiplicity constraints. The total count is always the product of the number of ways to choose the values and the number of ways to assign suits, divided by any overcounting due to indistinguishable selections. This structure holds because poker hands are defined by value groupings, and the suits are only relevant within those groupings.