Martin Gardner published the two-child problem in 1959. A family has two children. If you know that the older child is a boy, the chance that both are boys is one half. If you only know that at least one of them is a boy, the chance drops to one third. The two statements sound almost identical, which is why the result feels paradoxical.
The explanation is in the sample space. Two children can be boy-boy, boy-girl, girl-boy or girl-girl, each with probability one quarter. “The older one is a boy” leaves boy-boy and boy-girl, so half of the remaining cases are two boys. “At least one is a boy” removes only girl-girl, leaving three equally likely cases, one of which is two boys.
The answer also depends on how you learned the fact. If you meet one of the children at random and it is a boy, the answer is back to one half. Extra detail can matter too: knowing that one boy was born on a Tuesday raises the answer to 13/27. Choose a wording, enter your guess and the calculator shows the exact answer, the reasoning table and a simulation of thousands of random families.
How to use the Boy or Girl Paradox Calculator
- What do you know about the family?
- Your guess
- Chance that a child is a boy — 50% gives the textbook answers; the real figure is about 51.2%.
- Simulated families
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The calculator applies Bayes’ rule: P(both boys | statement) = P(both boys and statement) ÷ P(statement), with each child a boy with the chosen probability, independently. The simulation generates random families (sex and weekday of birth for each child), keeps those for which the statement is true and reports how many of them have two boys.
Worked example
“At least one child is a boy”: P(BB and at least one boy) = 1/4 and P(at least one boy) = 3/4, so P(BB | at least one boy) = (1/4) ÷ (3/4) = 1/3 ≈ 33.33%. For the Tuesday version list the 14 × 14 = 196 equally likely sex-and-weekday combinations: 27 contain a Tuesday boy and 13 of those have two boys, giving 13/27 ≈ 48.15%.
Answers for each wording
| What you know | Chance of two boys |
|---|---|
| The older child is a boy | 1/2 = 50% |
| At least one child is a boy | 1/3 ≈ 33.33% |
| At least one is a boy born on a Tuesday | 13/27 ≈ 48.15% |
| A randomly met child is a boy | 1/2 = 50% |
Formula
Older child is a boy: P(BB) = 1/2 · At least one is a boy: P(BB) = (1/4) ÷ (3/4) = 1/3 · At least one is a boy born on a Tuesday: 13/27
Worked example
Inputs — What do you know about the family?: At least one child is a boy; Your guess: 50%; Chance that a child is a boy: 50 %; Simulated families: 50,000.
Results — Chance that both children are of that sex = 33.33 %; As a fraction (50% boys) = 1/3; Your guess = Your guess of 50% is off – the answer is 33.33% (1/3).; Simulated frequency = 33.21 %; Which families fit the statement = Older, younger Prior chance Fits the statement
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Boy, Boy 25% yes
Boy, Girl 25% yes
Girl, Boy 25% yes
Girl, Girl 25% no.
Key terms
- Sample space
- The list of all possible outcomes – here BB, BG, GB and GG.
- Conditioning
- Restricting attention to the outcomes that agree with what you know.
- Independence
- The sex of one child does not change the probability for the other.
Frequently asked questions
If the older child is a boy, what is the chance that both are boys?
1/2. The younger child's sex is independent of the older child's.
If at least one child is a boy, what is the chance that both are boys?
1/3. Of the three equally likely families BB, BG and GB, only one has two boys.
Why do the two answers differ?
"At least one is a boy" does not say which child, so it only rules out GG. "The older one is a boy" rules out GB and GG.
What is the Tuesday boy problem?
If at least one child is a boy born on a Tuesday, the chance of two boys is 13/27 ≈ 48.1%. The extra detail makes the statement more specific, which pushes the answer towards 1/2.
Does the way I learned the information matter?
Yes. If you meet one child at random and it is a boy, the answer is 1/2, because a family with two boys is twice as likely to show you a boy.
Are boys and girls really equally likely?
Not quite – about 51.2% of births are boys. With that figure "at least one is a boy" gives about 34.4% instead of 33.3%.