Boy or Girl Two-Child Puzzle

This puzzle presents a simple two-child family scenario and asks for the probability that both children share a particular gender, given that at least one has that gender. It looks deceptively straightforward, making it a favorite in introductory probability interviews and brainteaser rounds. The key twist is that the extra piece of information about one child constrains the possible combinations of genders, and the question probes whether the candidate can correctly interpret this restriction rather than revert to naive intuition.

The solution relies on basic conditional probability applied to a small, discrete, equally likely sample space. A strong answer will systematically list all possible gender combinations, filter them using the stated condition, and then compute the desired probability as a ratio over this reduced set. Interviewers look for clarity in constructing the sample space, correct conditioning on the given information, and the ability to explain why some seemingly plausible outcomes are excluded. They also often pay attention to whether the candidate can articulate the difference between unconditional and conditional reasoning in everyday-language terms.

What it tests

When a problem involves conditional probability with equally likely discrete outcomes, the key is to enumerate the full sample space and then restrict attention to the subset consistent with the given condition. The probability of an event given some information is the ratio of the number of favorable outcomes to the number of possible outcomes that satisfy the condition. This approach works because, under independence and equal likelihood, each outcome is equally probable until additional information eliminates some. The principle is rooted in the definition of conditional probability: $P(A|B) = P(A \cap B)/P(B)$, and the intuition is that new information reshapes the set of possibilities, not their relative likelihoods.