In a knockout tennis tournament of 2^n contestants, the...

In a knockout tennis tournament of $2^{n}$ contestants, the players are paired and play a match. The losers depart, the remaining $2^{n-1}$ players are paired, and they play a match. This continues for $n$ rounds, after which a single player remains unbeaten and is declared the winner. Suppose that the contestants are numbered 1 through $2^{n}$, and that whenever two players contest a match, the lower numbered one wins with probability $p$. Also suppose that the pairings of the remaining players are always done at random so that all possible pairings for that round are equally likely. (a) What is the probability that player 1 wins the tournament? (b) What is the probability that player 2 wins the tournament?

Transcript

00:01 Once again, welcome to a new problem.

00:05 This time we're dealing with probability theory, and when it comes to probability, this is pretty much a chance process, where, for example, if you toss a coin, if you toss a coin, the probability of heads becomes one half, and the probability of tails also becomes one half.

00:32 So the purpose of probability is quantification of chances.

00:41 So you have different chances that happen to be quantified.

00:46 So we do have a new problem and this is a tennis tournament.

00:53 This is a tennis tournament and in the tennis tournament it involves a knockout.

01:01 So knockout means that you know once you play the contestants lose once you lose as a contestant you don't get a chance to play again so the number of number of contestants for this tournament happens to be two to the end power and players are paired so you have two players playing against each other the losers leave the tournament, the losers leave the tournament, and the players who remain, and in this case they're going to be two and minus one, are going to be paired too much.

02:13 Process this process is continuous this process is continuous for and rounds and the single remaining the single remaining player after the rounds is the winner.

02:52 The contestants, the contestants range from one up until two to the end.

03:08 And the lower, the lower numbered player wins by p probability.

03:30 Pairings for players are running.

03:37 And equally likely.

03:44 So they happen to be random and equally likely.

03:46 So there are two parts to this problem.

03:50 Determine probability that player one wins and then part b determine probability that player two wins.

04:17 So it wins the tournament.

04:20 So these are the probabilities we're looking at in the problem.

04:27 So we know, so we have the players.

04:36 The players happen to be one, two, three, all the way up until two to the end...

Question

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