A game shuffles 9 cards numbered 1 to 9 uniformly at random and lays them in positions 1 to 9. You are paid X² dollars, where X is the number of cards whose number equals their position. What is the expected payout in dollars?

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Squared matches payout

CombinatoricsHard

A game shuffles $9$ cards numbered $1$ to $9$ uniformly at random and lays them in positions $1$ to $9$ . You are paid $X^{2}$ dollars, where $X$ is the number of cards whose number equals their position. What is the expected payout in dollars?

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Hint

Expand $X^{2}$ using indicators and treat pairs of distinct positions separately.

Solution

Write $X = I_{1} + I_{2} + ... + I_{9}$ , where $I_{i}$ indicates that position $i$ holds card $i$ .
For each $i$ , $E (I_{i}) = \frac{1}{9}$ , and $I_{i}^{2} = I_{i}$ , so the diagonal terms contribute $9 \times \frac{1}{9} = 1$ .
For $i \neq = j$ , $E (I_{i} \times I_{j}) = \frac{1}{9 \times 8} = \frac{1}{72}$ , and there are $9 \times 8 = 72$ ordered pairs, so these contribute $72 \times \frac{1}{72} = 1$ .
Therefore $E (X^{2}) = 1 + 1 = 2$ .
The answer is $2$ .

Answer: $2$

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