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Eight tennis players take part in an elimination tournament and are placed at random. The players are ranked 1, 2, , 8, so the player ranked 1 is first (best) and the player ranked 8 is last (worst). When two players play a game, the higher-ranked player always wins. Thus, the player ranked first always makes it to the final round. Find the expected value of the rank of the other player in the final round. Also help with these! There are three cards in front of you, all face down. Each card has a real number written on it. You know that all three numbers are different, but you do not know what the numbers are.

You are allowed to choose a card and turn it over. At this point, you can either keep the card, or discard the card, and turn over a second card. Once you discard a card, you cannot return to it.

You have same option with the second card: You can either keep the second card, or discard it, and turn over the third card. If you discard the second card, your only option left is to turn over the third card and keep it.

At the end, if you have the card with the highest number on it, then you win a valuable prize. Otherwise, you leave empty-handed. If you follow the optimal strategy, then what is the probability that you win the prize?

Eight tennis players take part in an elimination tournament and are placed at random. The players are ranked 1, 2, , 8, so the player ranked 1 is first (best) and the player ranked 8 is last (worst). When two players play a game, the higher-ranked player always wins. Thus, the player ranked first always makes it to the final round. Find the expected value of the rank of the other player in the final round.             

Eight soccer teams participate in a soccer tournament, so that each team plays with every other team exactly once. If one teams wins against another, then the winning team gets two points, and the losing team gets zero points. If the two teams tie, then both teams get one point each.

After the tournament, the teams are ranked by the total number points each team won. What is the minimum number of points a team must win, to ensure that it is in the top four teams? Help!

 Jul 28, 2020
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For the card question: 

 

It turns out that there really isn't a good strategy; you should guess whichever card to be the card with the highest number on it, so the best you can do is 1/3.

 

Reference: https://math.stackexchange.com/questions/3474127/what-is-the-best-betting-strategy-to-maximize-probability-of-winning/

 Jul 29, 2020

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