The “boy or girl paradox” is a well-known brain teaser by famous puzzle-maker Martin Gardner. It’s popularly considered a “paradox” because (1) it has a highly unintuitive solution, and (2) its ambiguous wording meant either of two solutions could be valid solutions.

This is a rewording of that brain teaser to eliminate some ambiguity from that original question:

Out of all families with exactly two children, we randomly pick one family that has at least one boy. What is the probability that both children in this family are boys?

Assume only for the purposes of this puzzle that a child can only be a boy or a girl, and that either possibility is equally likely.

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