János and Gábor play the following game: János draws a card from a 32-card Hungarian deck. Gábor has to guess the color of the drawn card. If he guesses correctly, he wins 2 Ft from János, if not, he gives him 1 Ft. Before guessing, Gábor can name four cards, and János must tell him whether the drawn card is among them. How should Gábor choose the four cards to maximize his expected winnings?

Is the game fair? (A game is considered fair if the expected winnings of both players are $0 \mathrm{Ft}$.)
No matter which four cards Gábor names, the probability that the drawn card is among them is $\frac{4}{32}=\frac{1}{8}$, and the probability that it is not among them is $\frac{7}{8}$. Based on this observation, the game can be divided into two cases:

1. The drawn card is among the cards named by Gábor.

If Gábor names four cards of the same suit, he can know the suit of the card, so he can safely bet and win $2 \mathrm{Ft}$. If he does not name four cards of the same suit, he cannot be sure of the suit of the drawn card, so the expected value of his winnings is less than 2 Ft.

2. The drawn card is not among the named cards, but among the remaining 28 cards. In this case, it is advantageous for Gábor to bet on the color of which there are the most in the remaining 28 cards. There can be at most 8 cards of one color, so Gábor can guess the color of the drawn card with a probability of at most $\frac{8}{28}=\frac{2}{7}$, and with a probability of $\frac{5}{7}$ he will not guess correctly. Therefore, his expected winnings are at most $\frac{2}{7} \cdot 2 - \frac{5}{7} \cdot 1 = -\frac{1}{7} \mathrm{Ft}$.

The probability of the first case is $1 / 8$, and the probability of the second case is $7 / 8$, so Gábor's expected winnings are at most $1 / 8 \cdot 2 + 7 / 8 \cdot (-1 / 7) = 1 / 8$ Ft. If Gábor names four cards of the same suit (e.g., red) and then bets on red if the answer is "yes" and, for example, green if the answer is "no," he will fare the best in both cases. In any other choice, he will fare worse in either the first case, the second case, or both. The game is not fair: it favors Gábor.

Note. The game still favors Gábor even if he can only name three cards. However, if he can only name two cards, the game favors János. The game will be fair if, for example, Gábor can name eight cards, but his winnings for a correct guess are only $1 \mathrm{Ft}$ (and he still loses 1 Ft for an incorrect guess). It is interesting that if he can name 9 cards instead of 8, Gábor's chances do not improve.