15. Red and Black (from 8th grade, 3 points). The dealer ${ }^{2}$ deals one card at a time to the player from a well-shuffled standard deck of cards. At any moment, while the dealer still has cards left, the player can say "Stop". After this, the dealer reveals one more card. If it is of a red suit, the player wins; if it is black, the player loses.

Does there exist a strategy for the player that makes the probability of winning greater than $0.5 ?$[^6]
Solution. Suppose such a strategy exists, meaning that based on the sequence of already opened cards, the player can determine the optimal moment to say "Stop."

Let at this moment there remain $r$ red and $b$ black cards in the deck. Then the probability of success, i.e., the event "the next card is red," is $\frac{r}{r+b}$.

Change the rule and say "Stop" after the next card (let's call this rule "plus one"). Then there will remain $r+b-1$ cards, among which $r-1$ are red and $b$ are black with a probability of $\frac{r}{r+b}$, or $r$ are red and $b-1$ are black with a probability of $\frac{b}{r+b}$. Thus, with the "plus one" rule, the player will win with a probability of

$$
\frac{r}{r+b} \cdot \frac{r-1}{r+b-1} + \frac{b}{r+b} \cdot \frac{r}{r+b-1} = \frac{r(r+b-1)}{(r+b)(r+b-1)} = \frac{r}{r+b}
$$

In other words, the transition from the original strategy to the "plus one" strategy does not change the probability of winning. Reasoning similarly, we see that the probability will remain the same if the strategy is changed again and the "plus two" strategy is used. And so on. It turns out that if the player knows a winning rule, after which card to say "Stop," then they can say "Stop" not only after this card but also after any subsequent card, and the probability of winning will remain the same. Therefore, the rule "Stop after the second-to-last card" can be adopted regardless of the previous sequence of opened cards, and this will not change the probability of winning. This means that no matter what strategy is chosen, everything depends on the last card in the deck. But it can be black or red with equal probability. Therefore, no pre-chosen strategy can guarantee an advantage for the player before the game begins.

Answer: No.

Remark. What has been said does not mean that the probability cannot be greater than 0.5 at some point in the game. For example, if the deck has 36 cards, and at some point all 18 black cards have been opened, then all the remaining cards are red, and the player will win with a probability of 1. However, following the strategy "waiting for all black cards" can also lead to a guaranteed loss, specifically when, contrary to expectations, all red cards are opened first.