The key move is recognizing you don't need to think about the full distribution, just treat each of the 9 adjacent slots independently.
Use linearity of expectation by defining indicator variables for each adjacent position and summing their probabilities. For each of the 9 adjacent pairs, compute the probability that both cards are red or both are black using combinatorics, then add these probabilities together.
Pro tip: Mention that linearity of expectation holds even when events are dependent, which is why we can simply sum the probabilities without worrying about correlations between adjacent pairs.
For each adjacent position i (from 1 to 9), let X_i be 1 if the two cards at positions i and i+1 have the same color, and 0 otherwise. The total number of same-color adjacent pairs is the sum of X_i.
The expected value of the sum is the sum of the expected values: E[total] = sum_{i=1}^{9} E[X_i] = sum_{i=1}^{9} P(same color at positions i and i+1).
For any adjacent pair, the probability both are red is (6/10)*(5/9) = 30/90 = 1/3. Similarly, the probability both are black is (4/10)*(3/9) = 12/90 = 2/15. So P(same) = 1/3 + 2/15 = 5/15 + 2/15 = 7/15.
There are 9 adjacent pairs, so the expected number is 9 * (7/15) = 63/15 = 21/5 = 4.2.
Double-check the arithmetic and explain that the result is exact, not an approximation. Optionally, mention that the same method works for any counts of red and black cards.
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