Expected boys seated given a single-gender trio

Question

A group contains $3$ boys and $3$ girls. Three of the six children are picked at random to sit on three chairs. If you are told the three seated children are all the same gender, what is the expected number of boys among them?

Short answer

$3/2$ . With equal numbers of boys and girls, the all-boys and all-girls selections are equally likely (one way each), so the count is $3$ or $0$ with probability $\tfrac12$ each, giving $\tfrac32$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Conditional Expectation and Variance Revisited
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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