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Red neighbours in a deck

Expected valueMedium

A standard deck of $52$ cards, with $26$ red and $26$ black, is shuffled uniformly at random and laid out in a row. Count the adjacent pairs of positions where both cards are red. What is the expected value of this count? Give a fraction a/b in lowest terms.

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Hint

Find the probability that a specific adjacent pair of cards is entirely red, then multiply by the number of pairs.

Solution

The row has $51$ adjacent pairs of positions.
The first card of a pair is red with probability $\frac{26}{52}$ , and given that, the second is red with probability $\frac{25}{51}$ .
So a pair is double red with probability $\frac{26}{52} \times \frac{25}{51} = \frac{25}{102}$ .
By linearity the expectation is $51 \times \frac{25}{102}$ , which equals $\frac{25}{2}$ .

Answer: $\frac{25}{2}$

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