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If I flip a coin 9 times, what is the expected value of HT or TH in the run of 9 flips? Basically, I was thinking that HT or TH in the run of 9 flips would essentially mean the number of times that the coin changes from tails to heads or heads to tails. I was thinking of using $X$ as my random variable denoting number of times the coin changes sides, and $X_i$ as the random variable that is 1 if the $ith$ and $i + 1th$ flip values are different. However, I am not sure how to calculate the expected value of each $X_i$ and would appreciate some help in whether I am approaching this right.

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Your approach is sound. Each $X_i$ only depends on the $i^{th}$ flip and the $i+1^{st}$. They are all the same except for one. Now use the linearity of expectation to add them up for the total expectation.

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