Boy-girl neighbours in a random line of 25

Question

A class of $15$ boys and $10$ girls forms a single line in uniformly random order. Find the expected number of neighbouring positions occupied by one boy and one girl. (For example, the line $BGBBGGGBGGBBBBG$ has $7$ such boy–girl adjacencies.)

Short answer

$12$ . There are $24$ adjacent pairs of positions. Each is mixed with probability $2\cdot\tfrac{15}{25}\cdot\tfrac{10}{24}=\tfrac12$ , so by linearity the expected count is $24\cdot\tfrac12=12$ .

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

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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