How many cat-cat neighbors in a line of pets?

Question

Three dogs and five cats (8 pets total) are lined up for a photo in a single row in a completely random order. On average, how many of the 7 neighboring pairs consist of two cats?

Accepted Answer

A row of 8 pets has 7 neighboring pairs. Put one indicator on each pair. Fix one neighboring pair of positions. The two pets there are a random draw of 2 from the 8 pets without replacement. The chance both are cats is P(	ext{both cats}) = \frac58 \cdot \frac47 = \frac{20}{56} = \frac{5}{14}. Let Xi = 1 if neighboring pair i is two cats; E[Xi] = 	frac{5}{14}. With 7 pairs, E[X] = 7 	imes \frac{5}{14} = \frac{35}{14} = \frac52 = 2.5. Takeaway. For "how many neighboring pairs satisfy a condition," sum one indicator per gap. Each gap is two cats with probability 	frac{5}{14}, so the expected…

How many cat-cat neighbors in a line of pets?

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