A family has two children. A rule that truthfully reports whether at least one child is a boy born on a Tuesday returns yes. What is the probability both children are boys?

A family has two children. A rule that truthfully reports whether at least one child is a boy born on a Tuesday returns yes. What is the probability both children are boys?

Approach: Enumerate the 14 by 14 equally likely sex-and-weekday pairs, count how many contain a Tuesday boy, then count how many of those have two boys. Watch which sampling rule generated the statement.

13/27. Each child is one of 14 equally likely sex-and-weekday combinations, giving 196 equally likely ordered pairs. Counting the complement, the pairs with no Tuesday boy number 13^2 = 169, so 27 pairs qualify. Among the 49 two-boy pairs, 6^2 = 36 contain no Tuesday boy, leaving 13, so the conditional probability is 13/27 = 0.481. The weekday moves the answer from 1/3 towards 1/2 because it makes the two children distinguishable more often. The sampling rule is what fixes the number: had you met one child at random and seen a Tuesday boy, the answer would be 1/2.

Follow-up: How does the answer change if the reporting rule instead picks one child uniformly and reports that child's sex and weekday?

Key concepts: conditional probability, sampling rule, counting the complement, ordered pairs.