The puzzle
A parent says: “I have two children. At least one of them is a boy born on a Tuesday.” What is the probability that both children are boys?
The answer is 13/27, about 48.1 percent. Most people find this odd, because the day of the week seems to have nothing to do with it. Yet without the Tuesday detail, the answer would be 1/3. This puzzle is an extension of the two child problem, and it shows that extra detail can change a probability even when it looks irrelevant.
Assumptions
The answer relies on a standard set of simplifications:
- Each child is equally likely to be a boy or a girl.
- Each child is equally likely to be born on any of the seven days of the week.
- The two children are independent of each other, and the order (older, younger) is tracked.
- The statement is a fact about the family in the sense of “at least one is a boy born on a Tuesday”, not the answer to a question such as “is your first child a boy?”
Real births are not perfectly even across days and sexes, but the assumptions keep the arithmetic clean.
Building the sample space
Describe each child by two things: sex and birth day. That gives 2 × 7 = 14 equally likely types of child, such as “boy-Tuesday” or “girl-Friday”. For two children, ordered, there are 14 × 14 = 196 equally likely combinations.
Now count the combinations that fit the statement, “at least one child is a boy born on a Tuesday”. It is easier to count the opposite and subtract. A combination fails the statement if neither child is a Tuesday boy. Each child has 13 types that are not “boy-Tuesday” (14 minus 1), so there are 13 × 13 = 169 combinations where neither child is a Tuesday boy.
That leaves 196 − 169 = 27 combinations where at least one child is a Tuesday boy. These 27 are the cases that remain possible after hearing the statement, and each is equally likely.
Counting the two-boy cases
Among those 27, how many have two boys? Count the combinations with two boys and at least one Tuesday boy.
Two boys, ordered, with any birth days, gives 7 × 7 = 49 combinations. The ones with no Tuesday boy are those where neither boy was born on a Tuesday: 6 × 6 = 36. So 49 − 36 = 13 combinations have two boys with at least one born on a Tuesday.
The probability is therefore 13/27, about 0.481.
You can also check the 27 directly:
- Both children are Tuesday boys: 1 combination.
- Older is a Tuesday boy and younger is a boy born on another day: 6 combinations.
- Younger is a Tuesday boy and older is a boy born on another day: 6 combinations.
- Older is a Tuesday boy and younger is a girl (any day): 7 combinations.
- Younger is a Tuesday boy and older is a girl (any day): 7 combinations.
That adds to 1 + 6 + 6 + 7 + 7 = 27, and the first three groups, the two-boy cases, add to 13.
Why the weekday matters
Without the weekday, the statement “at least one is a boy” leaves three equally likely cases: boy-boy, boy-girl and girl-boy. Only one has two boys, so the answer is 1/3.
With the weekday, the statement becomes more specific. A family with two boys has two chances to contain a Tuesday boy, while a family with one boy has only one such chance. That makes two-boy families somewhat more likely to match the statement than they were before, so the probability rises from 1/3 towards 1/2. In the 27 combinations above, the double counting is visible: the two-boy families contribute 13 and the one-boy families 14.
As the detail gets more specific, such as a rarer day or a particular birth date, the answer moves closer to 1/2, because it becomes unlikely that both children match the detail, so the detail almost always refers to one child. With a very specific detail, the situation resembles “this particular child is a boy”, where the other child is a boy half the time.
Wording and how the information arrives
The result depends on how the parent came to say it. Here we treated the statement as a filter on all families: among all two-child families, we kept those that contain a Tuesday boy. Other readings give other answers. If the parent had picked one child at random and described that child, the probability of two boys would be 1/2. If the parent would always mention a boy when there is one, but would mention a girl otherwise, that again changes things.
This is why the puzzle is controversial in some discussions: it is easy to read the sentence differently. The two child problem and why wording matters is the heart of it. Probability answers depend on the process that produces the information, not just on the information itself. That is a key idea in conditional probability and in Bayes theorem, where evidence is weighed by how likely each possibility was to produce it.
Checking the figure
The counts above are small enough to verify with a spreadsheet simulation: generate two random children, each with a random sex and weekday, keep only the pairs that contain a Tuesday boy, and measure the fraction with two boys. With enough trials it should settle near 48 percent. Short runs will wander, so a few hundred trials will not be decisive.