Solution
A Boy Born on Tuesday
Show the problem again
A family is chosen uniformly at random from all families with exactly two children in which at least one child is a boy born on a Tuesday. Each child is independently a boy or a girl with probability 1/2, and is born on each of the seven days of the week with probability 1/7, independently of sex. What is the probability that both children are boys? Give your answer as a reduced fraction.
Worked solution
The answer is 13/27. Each child has 14 equally likely types (2 sexes times 7 days), so there are 196 equally likely ordered pairs. The pairs containing at least one Tuesday boy number 14 + 14 − 1 = 27, by inclusion–exclusion on "first child is a Tuesday boy" and "second child is a Tuesday boy". Among the two-boy pairs, the first is a Tuesday boy in 7 cases, the second in 7, and both in 1, giving 13. So the probability is 13/27. The day matters because the more specific the condition, the more nearly it singles out one child. "At least one boy" barely distinguishes between children and gives 1/3. "At least one Tuesday boy" nearly identifies a particular child and pushes the answer toward 1/2. Families with two boys get two chances to have a Tuesday boy, and the overlap counted only once is what keeps the answer below 1/2.
Source: Posed by Gary Foshee at the 2010 Gathering 4 Gardner; widely discussed since. Statement written for AxiomIQ.