Calling the next red card

Idea. The fraction of red cards left in the deck does not drift: on average it stays the same after each card. Any rule for when to speak is a stopping time, and stopping a quantity with no drift cannot beat its starting value, $\frac{1}{2}$.

Step 1: the fraction of red cards. With $r$ red and $b$ black cards left, speaking now wins with probability $X=\frac{r}{r+b}$. After the next card is turned, the fraction becomes $\frac{r-1}{r+b-1}$ with probability $\frac{r}{r+b}$ or $\frac{r}{r+b-1}$ with probability $\frac{b}{r+b}$. Its average is $$\begin{array}{rl}& \frac{r}{r+b}\cdot \frac{r-1}{r+b-1}+\frac{b}{r+b}\cdot \frac{r}{r+b-1}\\ & =\frac{r(r+b-1)}{(r+b)(r+b-1)}=\frac{r}{r+b}.\end{array}$$ So the fraction is a martingale.

Step 2: optional stopping. Whatever rule you use, you speak at a stopping time that is at most 51 cards in. For a martingale stopped at a bounded time, the mean is unchanged, so your chance of winning is $\mathbb{E}[X_{\tau }]=X_0=\frac{26}{52}=\frac{1}{2}$.

A slicker argument. Saying “the next card is red” wins exactly as often as saying “the last card is red”: given the cards turned so far, the next card and the last card have the same distribution. But a bet on the last card does not depend on when you make it, and the last card is red with probability $\frac{1}{2}$.

So no strategy beats $\frac{1}{2}$, and every strategy achieves it. Waiting for a favourable moment helps when it comes, but you may also never get one.