The added baby

EasyProbability Conditional Probability & Expectation~4m

A room has $5$ babies: $2$ boys and $3$ girls. One more baby, of unknown sex (equally likely boy or girl), is added to the room. A baby is then chosen uniformly at random from the $6$ and turns out to be a boy. Given this, what is the probability that the added baby was a boy?

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Split on the added baby's sex (prior $1/2$ each) and find $\mathbb{P}(\text{pick a boy})$ in each world.
Apply Bayes: posterior $\propto$ prior $\times$ likelihood. The two likelihoods are $1/2$ and $1/3$ .

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Asked at: Jane Street, Citadel

The added baby

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