Expected same-color neighbors in a shuffled deck

Question

You shuffle a standard 52-card deck (26 red, 26 black) and lay it out in a row. You then scan the 51 neighboring pairs of adjacent cards and count how many pairs are the same color. On average, how many same-color adjacent pairs are there?

Accepted Answer

Treat each of the 51 adjacent positions as its own little event and use linearity of expectation: the expected total is the sum of the per-pair probabilities. Focus on one adjacent pair. Whatever the left card's color is, the right card is one of the remaining 51 cards, of which 25 share that color. So P(	ext{a given pair is same-color}) = \frac{25}{51}. This is the same for every one of the 51 pairs by symmetry. Adding them up, E[	ext{same-color pairs}] = 51 	imes \frac{25}{51} = 25. The 51 in the denominator cancels the 51 pairs, leaving a clean 25. The pairs overlap and are strongly dep…

Expected same-color neighbors in a shuffled deck

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