Probability of Two Boys Given At Least One Son

This interview question presents two closely related child-gender scenarios that look similar on the surface but have different conditioning information. In one, you are told a mother of two qualifies for an event because she has at least one son. In the other, you literally observe one of a colleague's children and see that the specific child is a boy. The puzzle is to reconcile how these two forms of partial information affect the probability that both children are boys. This style of conditional probability question is common in quant trading, research, and data science interviews because it exposes whether a candidate treats probabilities intuitively or with precise definitions.

Solving it leans on constructing an appropriate sample space for two-child families, then carefully restricting and re-weighting that space under the different types of information. It tests understanding of conditional probability, symmetry, and how "at least one" information differs from observing a particular individual. Interviewers watch for rigorous enumeration of outcomes, clarity about assumptions, comfort with Bayes-type reasoning, and the ability to explain why the two parts yield different answers despite sounding deceptively similar.

What it tests

When dealing with probability problems involving families and conditional information (such as knowing at least one child has a certain property), the key is to carefully define the sample space and update it based on the information provided. The underlying structure is that the probability of an event changes when you condition on partial information, and the way you condition depends on whether the information is about the existence of a property ("at least one") or about a specific observed instance ("this child"). The principle is that the sample space must be restricted to only those outcomes consistent with the given information, and the probability is then computed relative to this reduced space. This pattern holds because conditional probability fundamentally re-weights the likelihood of outcomes based on what is known, and the way the information is presented (existential vs. specific) determines which outcomes remain possible and how they are counted.