At Least One Is a Boy
The problem
A family has two children. You learn that at least one of them is a boy. Assuming boys and girls are equally likely and independent, what is the probability that both children are boys? Then suppose instead that you meet one child at random and see that it is a boy: what is the probability now?
- Each child is independently a boy or a girl with probability .
- The information is genuinely only 'at least one is a boy' in the first case.
Take a minute
Reveal hint 1
Write down all four equally likely ordered outcomes first, then delete the ones the information rules out.
Reveal hint 2
In the second version, the way you obtained the information matters: meeting a child at random selects a child, not a family.
Solution
Answer for 'at least one is a boy'; when you meet a randomly chosen child who turns out to be a boy.
Derivation
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List the sample space.
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Condition on the first statement.
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Count the favourable case.
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Analyse the second version.
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Compute.
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Explain the difference.