The king is being randomly moved on the infinite chessboard. What is the expectation of modulus of difference between amount of white and black cells, which the king will visit by n steps. Each cell is counted as many times as king visits it.
Seemingly it is not that hard considering that the king has 8 cells to go (4 white, 4 black) and one can think that probabilities of coming to white and black cells are the same and expectation should equals to 0. But I feel that here is a catch and it's way more complex task
If $W$ denotes the number of visits to white cells then $W$ has binomial distribution with parameters $n$ and $p=\frac{1}{2}$.
To be found is: $$\mathbb{E}\left|W-\left(n-W\right)\right|=\mathbb{E}\left|2W-n\right|=2^{-n}\sum_{w=0}^{n}\left|2w-n\right|\binom{n}{w}$$
Can you do that yourself?