How many color blocks in a shuffled row of balls?

Question

Four red balls and four blue balls (8 total) are lined up in a random order. A "block" is a maximal run of the same color, so RRBRB has 4 blocks: RR, B, R, B.

On average, how many color blocks form in the row of 8?

Accepted Answer

The number of blocks equals 1 plus the number of neighboring pairs whose two colors differ. There are 7 neighboring pairs among 8 balls. Unlike coin flips, the colors here are drawn without replacement (fixed counts of 4 and 4). Fix a neighboring pair. The two balls differ when it is red-then-blue or blue-then-red: P(	ext{a given pair differs}) = \frac48\cdot\frac47 + \frac48\cdot\frac47 = 2 	imes \frac{16}{56} = \frac{32}{56} = \frac47. By linearity the expected number of switches is 7 	imes 	frac47 = 4, so E[	ext{blocks}] = 1 + 4 = 5. Takeaway. The "1 plus switches" trick works even fo…

How many color blocks in a shuffled row of balls?

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