Boy or Girl Paradox Calculator

The Boy or Girl Paradox (Two Children Problem)

The Boy or Girl paradox is a famous counterintuitive problem in conditional probability. It illustrates how subtle details in how information is specified drastically change probability outcomes.

The Three Classical Variations

1. At Least One Child is a Boy

For a two-child family, the equal-probability sample space of genders (order matters) is $\{BB, BG, GB, GG\}$. Knowing at least one child is a boy eliminates $GG$, leaving $\{BB, BG, GB\}$. Out of these 3 equally likely possibilities, only $1$ ($BB$) consists of two boys.

$$P(\text{Both Boys} \mid \text{At least 1 Boy}) = \frac{1}{3} \approx 33.33\%$$

2. The Older Child is a Boy

Knowing specifically that the older child is a boy restricts the sample space to $\{BB, BG\}$. Out of these 2 possibilities, $1$ ($BB$) consists of two boys.

$$P(\text{Both Boys} \mid \text{Older is Boy}) = \frac{1}{2} = 50\%$$

3. At Least One Boy Born on Tuesday (Gardner Variation)

When additional non-gender information (like day of birth) is attached to the specification, the probability shifts again. Taking 7 days in a week, there are $7 \times 2 = 14$ combinations per child. Calculating conditional probabilities yields:

$$P(\text{Both Boys} \mid \text{At least 1 Boy born on Tuesday}) = \frac{13}{27} \approx 48.15\%$$

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