11. The Ace of Diamonds (From 8th grade, 2 points). Thirty-six players are playing a game: from a deck of 36 cards, they take turns drawing a random card. If a player draws the ace of diamonds, the player wins; if another card is drawn, the player returns it to the deck, and the next player draws a card. They draw cards in a circle: first the first player, then the second, and so on. If no one draws the ace of diamonds in the first round, the players draw cards in the same order in the second round. This continues until someone draws the ace of diamonds.

Suppose that before the game starts, the players make bets, and the winner takes all. How should the players' bets relate to each other so that the game becomes harmless, that is, the mathematical expectations of the winnings for all players are equal to each other (and thus equal to zero considering the amount contributed)?
Solution. Let the probability of the first player winning be $p_{1}$. Suppose the first player does not draw the diamond ace on the first try. The probability of this is 35/36. At this point, the second player becomes the first, and the probability that he wins becomes equal to $p_{1}$. Returning to the beginning of the game, we see that for the second player to win, the first player must miss first. Therefore, at the start of the game, the probability of the second player winning is $\frac{35}{36} p_{1}$, which is less than $p_{1}$.

Similarly, the probability of winning for each subsequent player is the same multiple less than that of the previous one: $p_{k+1}=\frac{35}{36} p_{k}$. The game is unfair.

Let $L_{k}$ be the bet of player number $k$, and $L$ be the total bet, equal to the sum of the bets of all players. To make the game harmless, the expected value of the win for each player should be zero:

$$
\left(L-L_{k}\right) p_{k}+\left(1-p_{k}\right)\left(-L_{k}\right)=0, \text { from which } L_{k}=L p_{k}
$$

It turns out that the bets of the players should be proportional to the probabilities of their wins, that is, the bet of each subsequent player should be $\frac{36}{35}$ times smaller than the bet of the previous one.

Answer: The bet of each subsequent player should be in the ratio of 35:36 to the bet of the previous one.