Question 166
Manuel is a mathematician who likes card games. He meets brothers Jonas and Jonatan during a bus trip and proposes a game. Only the four aces of the deck will be used, the one with hearts and the gold ones are red while the one with swords and the one of clubs are black. 
Manuel will be the bench and the two brothers, one at a time, will bet 1 real against him in each round. The cards are placed face down. Jonas chooses a letter and Jonathan turns it up. Jonas chooses another letter and Jonathan turns her again. If both cards have the same color, then Jonas gets 1 Real from Manuel. Otherwise, Manuel wins 1 Real from Jonas. Then Jonas and Jonatan switch positions and the game continues. See that Manuel doesn't mess with the cards, so he can't manipulate the game. Jonatan thinks a little and concludes that he has a probability that $\frac{2}{3}$ is likely to win, as the results are just two red cards, two black or one red and one black. Really?
(a) Jonas has already participated in the Mathematics Olympics and decided to be more careful. He decided to analyze this game using a tree of possibilities. What would Jonah's possibilities tree look like?
(b) Considering the results of the tree from the previous item, what is the probability that Manuel will win each round of the game?
Manuel will be the bench and the two brothers, one at a time, will bet 1 real against him in each round. The cards are placed face down. Jonas chooses a letter and Jonathan turns it up. Jonas chooses another letter and Jonathan turns her again. If both cards have the same color, then Jonas gets 1 Real from Manuel. Otherwise, Manuel wins 1 Real from Jonas. Then Jonas and Jonatan switch positions and the game continues. See that Manuel doesn't mess with the cards, so he can't manipulate the game. Jonatan thinks a little and concludes that he has a probability that $\frac{2}{3}$ is likely to win, as the results are just two red cards, two black or one red and one black. Really? (a) Jonas has already participated in the Mathematics Olympics and decided to be more careful. He decided to analyze this game using a tree of possibilities. What would Jonah's possibilities tree look like?
(b) Considering the results of the tree from the previous item, what is the probability that Manuel will win each round of the game?