Hi there, here's another puzzle I've been looking at.

Suppose you are to guess the colour of the next card in an ordinary deck of 52 cards---red or black---one at a time. How many can you expect to get right?

Here's what I think. Let $E(r,b)$ denote the expected number you will get right given that there are $r$ red cards left and $b$ black cards left. Then

If $r \geq b$, then you would guess red (WLOG in the case of equality) and thus $$E(r,b) = \frac{r(1+E(r-1,b))+ b E(r,b-1)}{r+b}.$$

If $r < b$, then you would guess blue and thus $$E(r,b) = \frac{r E(r-1,b) + b(1+E(r,b-1))}{r+b}.$$

Thus, if we let $$F(r,b) = \frac{r E(r-1,b) + b E(r,b-1)}{r+b},$$ then for all $r,b$ we have $$E(r,b) = F(r,b) + \frac{max(r,b)}{r+b}.$$

How might one solve these equations? Simulation tells me that $E(26,26) = 30.0392...$.