Probability in a sport tournament

[Sleek]

Homework Statement

There are 8 athletes, each numbered from 1 to 8. They all play a game against each other. Thus in each round, a group of two is randomly made. In a group, a person with lower number always wins. eg, a match between 3 and 7 would be always won by 3. The losers are eliminated. Thus in the second round, only 4 are present, and in the third, 2. Find the probability that athlete 4 reaches the finals

Homework Equations

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The Attempt at a Solution

Athlete 1 will always be victorious, because he cannot be defeated by others. But in the finals, he'd face a person, and I have to find the probability that he is athlete 4.

For no. 4 to go to the second round,

Lets say sample space is 8C2. No. of combinations that'd lead him to win is 4-5,4-6,4-7,4-8, i.e. 4 combinations.

His probability of going to second round is 1/7.

In the next round, along with number 4, there are two possibilities. Let's consider all no. below 4 to be X and above 4 to be Y.

In the second round, athletes can be present in such order,

4,X,Y,Y --- 4,X,X,Y [Since atleast one X would always be present (1). If all the three are X, probability of 4 going to finals becomes 0]

In the first case, probability of 4 winning is 2/4C2=1/3 and in second, 1/6.

So probability of 4 going to finals is 1/7*(1/6+1/3)=1/14. But according to the book, this is wrong since the answer given is 4/35. Can someone help?

 

[Avodyne]

In the first round, the number of possible pairings is not 8C2, but rather 7*5*3*1. To see this, ask, who is #1 paired with? There are 7 possibilities. Once #1's opponent has been chosen, pick a player who's left (#2, say, if #2 is not #1's opponent; otherwise, #3), and ask who that player's opponent is. There are 5 possibilities. And so on.

Now, in that first round, what are the odds that #4 draws #5, 6, 7, or 8, and not #1, 2, or 3?

Say #4 draws #5. To survive the next round, we must not have #1, 2, and 3 paired up with #6, 7, and 8. What are the odds that this does NOT happen?

 

[Architeuthis Dux]

I would suggest approaching this problem using a different approach, such as using a probability tree diagram or a Markov chain. This will help visualize the different outcomes and make it easier to calculate the probabilities.

Additionally, the given information about the game being played between two athletes at a time can be interpreted as a round-robin tournament, where each athlete plays against every other athlete once. In this case, the probability of athlete 4 reaching the finals can be calculated by considering the different paths that lead to the finals.

For example, in the first round, athlete 4 can either win or lose against athlete 1, and the same goes for each subsequent round. This results in a total of 8 possible paths that lead to the finals, with 4 of them leading to athlete 4 winning and 4 leading to athlete 4 losing. Therefore, the probability of athlete 4 reaching the finals is 4/8 or 1/2.

I would also suggest checking the given answer of 4/35 to ensure that it is correct, as it may be a typographical error or a different interpretation of the problem.