Boy or Girl Paradox Calculator — Two-Child Problem
A family has two children. Knowing that at least one is a boy — or that the older one is, or that one was born on a Tuesday — gives surprisingly different answers for the probability that both are boys. Select a scenario to see the conditional probability and why it is counterintuitive.
What is known about the family?
1 / 3 ≈ 0.3333
- 1
Sample space size (after condition)
3at least one boy (GG excluded) - 2
Favourable outcomes (both boys)
1 - 3
P(both boys | condition)
1 ÷ 3 = 0.3333
How does this calculator work?
A two-child family with "at least one boy" has P(both boys) = 1/3, not 1/2, because the sample space {BB, BG, GB} has three cases with only one favourable. Knowing the specific child (older is boy) gives 1/2; adding a day-of-birth constraint (Tuesday boy) gives 13/27 ≈ 0.481 — all from the same conditional-probability rule P(A|B) = P(A∩B)/P(B).
Formula
How this is calculated
The two-child problem is a classic exercise in conditional probability. The full sample space for two children (treating birth order as fixed) is {BB, BG, GB, GG} — four equally likely pairs. The probability of "both boys" in this unrestricted space is 1/4.
When told "at least one is a boy," GG is eliminated, leaving {BB, BG, GB} — three equally likely outcomes, only one of which is BB. Therefore P(BB | at least one boy) = 1/3. This surprises many people who expect 1/2 because "the other child is either a boy or a girl."
When told "the older child is a boy," only BB and BG are possible — so P(BB | older is boy) = 1/2. Knowing *which* child is the boy, rather than merely that at least one is, halves the sample space. This shows that the wording of the condition dramatically changes the probability.
The "Tuesday boy" version illustrates that adding even more specific information shifts the probability further toward 1/2. With a 7-day week, 27 ordered pairs satisfy "at least one boy born on a Tuesday," and 13 of those are both-boys, giving P = 13/27 ≈ 0.481. The paradox is real and counterintuitive — the result has been verified formally and appears in peer-reviewed puzzle literature.
Frequently asked questions
"At least one boy" eliminates only GG, leaving {BB, BG, GB}. Because BG and GB are different ordered outcomes (older-younger), there are two ways to have exactly one boy and only one way to have two boys. Hence 1/3, not 1/2. If the two children were indistinguishable you would get 1/2, but birth order makes them distinct.
Specifying which child is a boy fixes one child's sex entirely, leaving the other child's sex as the only unknown. Now only two outcomes are possible — BB or BG — and each is equally likely, giving a straightforward 1/2.
Additional specific information reduces the size of the "other family" sample space less quickly than it reduces the "both boys" count, shifting the ratio toward 1/2. As the condition narrows to one specific event (a particular birthday, for example), the probability of both boys approaches but never reaches 1/2 via this kind of constraint.
Also known as
TG we-Calculate Editorial Team. (2026). Boy or Girl Paradox Calculator — Two-Child Problem [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/boy-or-girl-paradox-calculator
TG we-Calculate Editorial Team. "Boy or Girl Paradox Calculator — Two-Child Problem." TG we-Calculate. 2026. https://we-calculate.com/calculator/boy-or-girl-paradox-calculator.
TG we-Calculate Editorial Team, "Boy or Girl Paradox Calculator — Two-Child Problem," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/boy-or-girl-paradox-calculator
@misc{wecalculate_boy_or_girl_paradox_calculator, title = {Boy or Girl Paradox Calculator — Two-Child Problem}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/boy-or-girl-paradox-calculator}}, year = {2026}, note = {TG we-Calculate} }
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