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Colour runs in a row of beads

Expected valueMedium

$3$ red beads and $3$ blue beads are arranged in a line, with all arrangements equally likely. A run is a maximal block of same-coloured neighbours. What is the expected number of runs?

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Hint

Count adjacent pairs with different colours.

Solution

There are $5$ adjacent pairs, and a pair has different colours with probability $2 \times 3 \times \frac{3}{6 \times 5} = \frac{3}{5}$ .
The number of runs is $1$ plus the number of colour changes.
The expectation is $1 + 5 \times \frac{3}{5} = 4$ .

Answer: $4$

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