A family has two children. You know at least one of them is a boy. What is the chance both are boys? Most people say 1/2; the answer is 1/3. Add that the boy was born on a Tuesday, and the answer becomes 13/27. Let's see why a weekday changes anything, and why the answer depends on how you learned about the boy.
The two-child problem, also called the boy or girl paradox, was published by Martin Gardner in 1959. People still argue about it, because the same sentence can be read in different ways.
The neighbours have two children. Each child, independently of the other, is a boy or a girl with probability 1/2.
Writing the older child first, there are four options: BB, BG, GB, GG. Each one is a quarter of all families.
Version A: "at least one child is a boy". Version B: "the older child is a boy". Version C: "at least one child is a boy born on a Tuesday".
In each version, what is the chance that the family has two boys?
"The other child is either a boy or a girl. So it's always 1/2"
In fact the answers differ: 1/3, 1/2 and 13/27. "At least one" does not say which child is the boy, so it only rules out the GG family. And a weekday, oddly, pushes the answer towards 1/2.
The last assumption matters most. If the information was obtained differently, the answer changes. More in round 4.
For each sentence, estimate the chance that the family has two boys. Then press "Check".
Your estimates will be compared with the right answers here.
Each card is a random family: the older child on the left, the younger on the right, each child marked with the weekday of birth. Pick a sentence, and the families it is false for fade out. Count what share of the rest have two boys.
| Sentence | Two boys / kept | Theory |
|---|
In a small deck the shares jump around. Add a few hundred families and they settle near the theory. The Tuesday sentence keeps only about 14% of the deck, so its spread is the widest.
The computer builds 100,000 random families, keeps the ones each sentence is true for, and counts the share with two boys. The dark tick marks the exact answer.
The sentence "at least one child is a boy with some detail", where each child has the detail with chance 1 in N. The line is the exact formula; the dots are simulations, each running through millions of families until several thousand are kept.
One question decides everything: which families remain once you have heard the sentence. Here are four ways to see it, plus the formulas.
In the second case a BB family always shows you a boy, while BG and GB do so only half the time. Weights 1, ½, ½ give BB a share of exactly 1/2. Both parents could say "we have a boy", but the procedure that led to the sentence is different.
Rows: the older child. Columns: the younger. A square is a boy, a circle a girl, the letters are weekdays. All 196 cells are equally likely.
In the Monty Hall problem the answer 2/3 holds only because the host knows where the car is and always opens a goat door. If the host opens a door at random and happens to show a goat, the chance becomes 1/2. The same here: if you selected every family with at least one son, the answer is 1/3. If a boy just happened to cross your path, it is 1/2.
And why does Tuesday matter? A family with two boys has two chances to have a Tuesday boy, a family with one boy has only one. So BB passes the filter almost twice as often, and its share rises from 1/3 to almost 1/2.
The numerator ¼·(2p − p²) is the chance that in a BB family at least one of the two boys has the detail. The term ½·p covers BG and GB, where the only boy must have it. When p is small, 2p − p² ≈ 2p, so BB weighs ½·p, exactly as much as BG and GB together.
Any detail about the boy works like Tuesday: the rarer the detail, the closer the answer to 1/2. Try it, then see what changes with three children and with the real sex ratio.
The chance that all three are boys. With no information it is 1/8. "Check" simulates 100,000 families.
| What you know | Exact |
|---|
About 105 boys are born for every 100 girls, so the chance of a boy is ≈ 51.2%. Switch the model and see how much the answers move.
| What you know | 50 : 50 | 105 : 100 | Chosen model |
|---|
Martin Gardner posed both questions, about the older child and about "at least one boy", in his Mathematical Games column in Scientific American in 1959. He gave the answers 1/2 and 1/3, and later admitted the second question was ambiguous.
The Tuesday version was posed by Gary Foshee at the Gathering 4 Gardner conference in 2010, held in Gardner's honour. Afterwards it spread quickly through newspapers and blogs and became one of the most argued-about puzzles.
Of the 196 equally likely "sex and weekday" combinations for two children, the Tuesday-boy sentence is true for only 27. The family where both sons were born on a Tuesday counts once, which is why it is 13 of 27 and not 14 of 28.
Worldwide, about 105 boys are born for every 100 girls. For our puzzle this changes almost nothing: the "at least one boy" answer goes from 33.3% to 34.4%.
Maya Bar-Hillel and Ruma Falk, in "Some teasers concerning conditional probabilities" (Cognition, 1982), showed that the answer to puzzles of this kind depends on how the information was obtained, not only on what it says.
The Parade columnist who explained the Monty Hall problem discussed versions of this puzzle in 1991 and 1996. When readers questioned her analysis, she surveyed readers with exactly two children, at least one of them a boy. Of 17,946 responses, 35.9% reported two boys, close to 1/3.
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