Suppose we have 52 decks with 26 red and 26 black cards. We shuffle them at a random order. Then we define a "block" as cards with same colors, for example, BRRB has 3 blocks and BRRRBBRRRR has 4 blocks. So how many blocks we would have, in the sense of expectation?
My though is that, we set indicator function $I_i$ representing the i-th card has different colors with the card on the left of it. So $I_i=1$ if it is different, and is zero if it's same. So the final number of blocks would be equal to $\sum_{i=1}^{52}I_i$, and $E\sum_{i=1}^{52}I_i = \sum_{i=1}^{52} P(i)$, where P(i) is the probability that i-th card has different color comparing to his left one. But I got stuck at this step cuz I don't know how to calculate the probability of $i-th$ term. Is there anybody who can help me?
