Dice sequence: how many neighboring pairs match?

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You roll a fair six-sided die 12 times and write the results in a row. On average, how many of the 11 neighboring pairs show the same value on both rolls?

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Twelve rolls make 11 neighboring pairs. Put one indicator on each pair and add. Fix a neighboring pair of rolls. The second roll matches the first only if it lands on that exact face, which happens with probability 	frac16: P(	ext{a given pair matches}) = \frac16. Let Xi = 1 if neighboring pair i matches; E[Xi] = 	frac16. With 11 pairs, E[	ext{matching pairs}] = 11 	imes \frac16 = \frac{11}{6} \approx 1.833. Takeaway. The condition on each neighboring pair sets its probability: matching faces on two dice happen with probability 	frac16, so 11 pairs average 	frac{11}{6} matches. Lineari…

Dice sequence: how many neighboring pairs match?

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