Revised Gender Paradox with a Daughter

This question revisits the classic "at least one girl" puzzle, but changes how you learn the information. Instead of only being told something about the pair of children as a group, you now meet and identify a specific daughter at the door. The problem asks for the probability that both children are girls given this more concrete, individual-level observation, and then asks you to contrast it with the answer to the earlier, purely verbal version. The focus is on how the probability changes when you move from an abstract group condition to seeing an actual child.

The solution leans on conditional probability and on defining the correct sample space under different information protocols. Candidates must clearly distinguish between events about "at least one child" and events about "this particular child," and model how conditioning changes when a specific individual is singled out. Interviewers watch for careful enumeration of cases, consistent use of Bayes-style reasoning, and an explicit discussion of why the result differs from, or coincides with, the earlier version of the problem. They look for precision in specifying assumptions about how the observation arises.

What it tests

When conditioning on observed information about individuals in a group, the way information is acquired fundamentally alters the underlying probability space. In problems involving independent random variables (like children's genders), being told a fact about the group (such as 'at least one is a girl') restricts the sample space by eliminating some outcomes, but does not distinguish between individuals. However, observing a specific individual (such as meeting a girl at the door) singles out one random variable, making the remaining uncertainty about the others independent of the observation. This difference arises because direct observation gives you information about a specific instance, while a group-level statement only constrains the set of possibilities. The principle is that the method of acquiring information (group-level vs. individual-level) determines how you update probabilities and which outcomes remain possible.