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We observe a tournament of 2 ^ n players, which consists of n rounds. In each round every 2 players play against each other and only the winner goes to the next round. At the beginning of each round random pairings are defined, all with the same probability.

Each player is assigned a number corresponding to its position in a ranking, ie if [2 ^ n] is the set of all participants, player i is worse than player j if i> j (1st place, 2nd place, ...)

The encounters of a round are played in the following order: of all remaining encounters to be played, the next encounter will count with the 2 worst players. In each of the encounters of 2 players the best of both always wins with a probability p∊(0.1) - regardless of all other ecounters of the tournament

(a) Is n = 2

Draw a tree diagram of the experiment above exposed calculating the probabilities depended on p where:

i) player 1 wins

Solution?:

If n = 2 then there are 2 ^ 2 = 4 players. I have to find all possible combinations between players A = 1, B = 2, C = 3 and D = 4 but considering that the round (A versus B) and (C versus D) ist the same as (B, A) and (C, D) or (C, D) and (A, B). So I concluded that there are only three possible events:

(A, B) and (C, D)

(A, C) and (B, D)

(A, D) and (B, C)

However, the statement says "The encounters of a round are played in the following order: of all remaining encounters to be played, the next encounter will count with the 2 worst players."

Does this mean then for n = 2 that the only possible case is ((A, B) and (C, D)) because the two worst are C and D because their corresponding ranking numbers are 3 and 4 respectively? If it is so, would be the tree like this?:

Therefore there would be two possible cases where A (Player 1) wins,:

A beats B and C beats D. Then A beats C

A beats B and D beats C. Then A beats D

The problem is that I do not know how to calculate the probabilities of these events.

Can someone help me? Conditional probability should be used?

ii) player 2 plays in the final round with player 2

iii) player 1 plays at some point with player 2

(b) Let n be any number

i) How many random events Ω kε (branches) have?

ii) Calculate the probabilities for the events E ⊆ Ω of (a) for any n

I'll be glad if somebody helps me =)