problem 41

4.3.4 problem 41

Let $I_{j}$ be the indicator variable for $j$ -th card being red. Let $R_{j} = I_{j} I_{j + 1}$ be the indicator variable for the $j$ -th and $j + 1$ -st cards being red. Let $X = \sum_{j = 1}^{51} R_{j}$ be the number of consecutive red pairs in a well shuffled deck of 52 cards. Then,

E (X) = E (51 \sum j = 1 R_{j}) = 51 \sum j = 1 E (R_{j}) = 51 \sum j = 1 \frac{(\frac{26}{2})}{(\frac{52}{2})} = 51 \frac{(\frac{26}{2})}{(\frac{52}{2})}

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