How many blocks of equal values in six dice rolls?

Question

You roll a fair six-sided die 6 times in a row. A "block" is a maximal run of equal consecutive values, so 2,2,5,5,5,1 has 3 blocks: 2s, 5s, and 1. On average, how many blocks appear in the 6 rolls?

Accepted Answer

The number of blocks equals 1 plus the number of neighboring pairs whose two values differ. There are 5 neighboring pairs among 6 rolls. Fix a neighboring pair. The second roll differs from the first unless it lands on the same face, which happens with probability 	frac16, so P(	ext{a given pair differs}) = 1 - \frac16 = \frac56. By linearity the expected number of switches is 5 	imes 	frac56 = 	frac{25}{6}, so E[	ext{blocks}] = 1 + \frac{25}{6} = \frac{31}{6} \approx 5.167. Takeaway. The same recipe handles dice: neighboring rolls differ with probability 	frac56, so the expected numbe…

How many blocks of equal values in six dice rolls?

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