Four-of-a-Kind from Three-of-a-Kind

This classic cards problem is about conditional probability in a finite combinatorial sample space. You already know that a certain structure is present in the hand, and you must update the probability of a more specific structure under that information. It's a clean example of how quant interview questions translate verbal constraints into precise mathematical events.

Solving it in quant prep trains your ability to model "given that" information rigorously and to enumerate discrete outcomes under constraints. It reinforces comfort with counting techniques and with translating English descriptions of cards and ranks into events, intersections, and new sample spaces.

For quant interviews, this matters because trading, risk, and derivatives problems often come with partial information. Interviewers want to see you update probabilities correctly when the universe of outcomes is restricted by what you already know.

What it tests

When a problem asks for the probability of a more restrictive event given a broader event, the core structure is conditional probability, where the denominator counts all cases satisfying the broader condition and the numerator counts only those that also satisfy the stricter condition. In combinatorial settings, this often means partitioning the sample space into mutually exclusive cases that exhaust all possibilities under the given constraint. The key is to recognize that the conditioning changes the 'universe' of possible outcomes, so you must count only those hands that meet the stated minimum requirement, not all possible hands. This approach generalizes: whenever a minimum or 'at least' condition is given, enumerate all ways to meet that minimum, then further restrict for the event of interest. The principle holds because conditional probability always rescales the sample space to the set of outcomes consistent with the given information, and combinatorial enumeration is the tool for precise counting within these sets.