Manuel is a mathematician who enjoys card games. He meets the brothers Jonas and Jonatan during a bus trip and proposes a game. Only the four aces from the deck will be used, the ace of hearts and the ace of diamonds are red, while the ace of spades and the ace of clubs are black.

![](https://cdn.mathpix.com/cropped/2024_05_01_40ea5c1759d4b041c087g-15.jpg?height=237&width=765&top_left_y=2194&top_left_x=748)

Manuel will be the banker, and the two brothers will bet 1 real against him in each round, one at a time. The cards are placed face down. Jonas chooses a card, and Jonatan turns it over. Jonas chooses another card, and Jonatan turns it over again. If the two cards have the same color, Jonas wins 1 real from Manuel. Otherwise, Manuel wins 1 real from Jonas. Then Jonas and Jonatan switch positions, and the game continues. Note that Manuel does not touch the cards, so he cannot manipulate the game. Jonatan thinks for a moment and concludes that he has a $\frac{2}{3}$ probability of winning, as the outcomes are only two red cards, two black cards, or one red and one black. Is that really the case?

(a) Jonas has participated in math olympiads and decided to be more careful. He decided to analyze this game using a probability tree. What would Jonas's probability tree look like?

(b) Considering the results from the tree in the previous item, what is the probability of Manuel winning each round of the game?
Solution

(a) We will represent clubs, spades, diamonds, and hearts by the letters $C, S, D$, and $H$, respectively. The tree of possibilities is shown in the figure below:

![](https://cdn.mathpix.com/cropped/2024_05_01_40ea5c1759d4b041c087g-16.jpg?height=831&width=425&top_left_y=1184&top_left_x=724)

(b) Using the tree of possibilities, there are 12 possible outcomes, and in 8 of them, Manuel wins. Therefore, the probability of Manuel winning each round is $\frac{8}{12}=\frac{2}{3}$. Jonatan's reasoning is not correct, as the possibilities of colors do not occur with the same probability.

It is worth noting that this item could be answered without the tree of possibilities. Consider the moment after the first card is turned. Among the other three, two favor Manuel, as they have a different color from the card that was turned. Thus, as in the conclusion using the tree, the probability of Manuel winning is $\frac{2}{3}$.