Math Is Fun Forum

mortage762

Member

Registered: 2007-05-04

Posts: 6

Probability question

This question was confusing me so any help would be appreciated!

Suppose that there is a tennis tournament with 32 players. Players are matched in a completely random manner, and we assume that each player always has probability 1/2 of winning a match. What is the probability that two given players meet each other during the tournament?

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Probability question

Hmm, that's quite a tricky one. There might be an easier way, but here's how I'd do it:

First we need to work out the probabilities for them being certain distances away from each other in the drawing.

Of the half of the draw that player A is in, there are 15 other players joining him. There are 16 spots in the other half, and so the probability that the two given players will meet in the final (assuming they always win) is 16/31.

Similarly, there's an 8/31 chance of them meeting in the semis, 4/31 of meeting in the quarters, 2/31 of them meeting in the second round and 1/31 of them meeting in the first.

Naturally, if the draw matches them against each other in the first round then they're bound to face each other because they've had no opportunity to be knocked out at that stage.

However, by the second round there's only a 1/4 chance that they're both still in play, because they both had a 1/2 chance of being beaten.

Similarly, there's a 1/16 chance that they've survived up to the quarters, 1/64 that they're in the semis, and 1/256 that they're in the final.

That means that the probabilities of them meeting in each round is:

1st: 1/31
2nd: 1/4 x 2/31 = 1/62
Quarters: 1/16 x 4/31 = 1/124
Semis: 1/64 x 8/31 = 1/248
Final: 1/256 x 16/31 = 1/496

Totalling all of those up gives a final probability of 31/496 that they will meet at some point during the tournament.

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Why did the vector cross the road?
It wanted to be normal.