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Juniors side by side

Expected valueEasy

$10$ traders sit in random order around a round table, and $3$ of them are juniors. What is the expected number of juniors whose left-hand neighbour is also a junior? Give a fraction in lowest terms.

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Hint

Use one indicator for each junior.

Solution

For a given junior, the left-hand neighbour is equally likely to be any of the other $9$ traders, of whom $2$ are juniors, so the chance is $\frac{2}{9}$ .
By linearity, which needs no independence, the expected number is $3 \times \frac{2}{9} = \frac{2}{3}$ .

Answer: $\frac{2}{3}$

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