An urn contains $n$ white and $n$ black balls. We draw them without replacement, but before doing so, we must predict the outcomes of all the draws. What is the expected number of correct predictions?
We denote the white balls with $H$ and the black ones with $K$. The total number of different outcomes of the drawing is $\binom{2 n}{n}$, and since the draws are independent of each other, each outcome is equally likely. Let's calculate how many hits we can achieve with a given tip in all the draws. For each draw, we associate a sequence where every $H$ is replaced by $K$ and vice versa (for example, $H H K H K K-$ is associated with the sequence $K K H K H H$). In these two sequences, we will have exactly $2 n$ hits in total, since any tip that hits in the first sequence will miss in the second, and vice versa. Therefore, the total number of our hits is $n\binom{2 n}{n}$, and on average, we get $n$ hits per draw. Thus, the expected value of the number of hits is $n$.