Boy or Girl Paradox Calculator
The probability that both children are boys depends entirely on what information you know. Select different scenarios to see how the answer changes.
Given that a two-child family has at least one boy, what is the probability that both children are boys?
Enter a value between 0 and 1. The classic paradox uses p = 0.5 but you can adjust it.
Probability Both Are Boys
33.33%
At Least One Child is a Boy
Numerator
0.2500
Favorable probability weight
Denominator
0.7500
Conditioned probability mass
| Family Combination | Probability |
|---|---|
| Boy / Boy | 0.2500 |
| Boy / Girl | 0.2500 |
| Girl / Boy | 0.2500 |
| Girl / Girl | 0.2500 |
How to Use This Calculator
- Select the conditional information you know about the family.
- Adjust the base probability that any child is a boy (default 50%).
- Read the resulting probability that both children are boys under your chosen condition.
- Examine the probability table of family combinations to confirm the calculation.
Formula
For the “at least one boy” scenario:
P(BB | ≥1B) = P(BB) / [1 − P(GG)] = p² / (1 − (1 − p)²)
For the “older child is a boy” scenario:
P(BB | oldest boy) = P(youngest boy) = p
These formulas highlight that the paradox is not about arithmetic but about conditioning on the right sample space.
Full Description
The boy-or-girl paradox demonstrates how ambiguous phrasing can lead to seemingly contradictory answers. If you only know that at least one child is a boy, the possible families are BB, BG, and GB. Only one of the three has two boys, so the probability is 1/3. If instead you know the older child is a boy, there are just two options (BB and BG), yielding a probability of 1/2.
The key idea is that conditioning filters the sample space. Different information removes different family configurations, producing distinct results. Always clarify what information is given before applying probability rules.
Frequently Asked Questions
Why can the probability be 1/3 or 1/2?
Because the conditional information differs. “At least one boy” keeps three equally likely family types, whereas “older child is a boy” keeps only two. The paradox disappears once the conditioning event is stated clearly.
What if the children are not equally likely to be boys?
Set p to any value between 0 and 1. The calculator adjusts both scenarios accordingly, illustrating how asymmetry changes the probabilities.
Does the order of children matter?
Yes. The “older child” scenario explicitly uses order, while the “at least one boy” scenario does not. Accounting for order doubles certain outcomes, which is crucial for the correct answer.
Can this paradox involve girls instead?
Absolutely. Swapping “boy” for “girl” yields identical logic. The paradox stems from conditioning, not gender.