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Topic: Anyone For Tennis? (Read 9285 times) |
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SWF
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Re: Anyone For Tennis?
« Reply #26 on: Sep 8th, 2007, 6:36pm » |
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Instead of using the infinite series that Eigenray gives for the win by 2 rule, the way I would do it is let r be probability that the server wins by 2 after the score is tied at 3-3 or more. r = p*p + 2*p*q*r (probability of server winning two straight points plus probability of the score being tied after the next 2 games and server wins after that). Solving for r gives, r=p*p/(1-2*p*q).
Similar to what Eigenray said, add probablity of the four cases: server wins 4-0, 4-1, 4-2, or after score has reached 3-3:
p^4 + 4*p^4*q + 10*p^4*q^2 + 20*p^3*q^3*( p*p/(1-2*p*q) )
The last two terms can be combined to give the 10*p^4*q^2/(1-2pq) appearing in the solutions given previously.
You could continue with a similar analysis of probability of winning a set, and enough sets to win a match, but would need to have two probabilities depending on if a given player is serving or receiving. What is better for determining the better player: the current format of game, set, match or just playing points until one of the players reaches a certain number of points first?
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srn437
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Re: Anyone For Tennis?
« Reply #27 on: Sep 8th, 2007, 6:52pm » |
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Ok.
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #28 on: Sep 8th, 2007, 7:42pm » |
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on Sep 8th, 2007, 6:36pm, SWF wrote:| What is better for determining the better player: the current format of game, set, match or just playing points until one of the players reaches a certain number of points first? |
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The rules of table-tennis were changed a few years ago. One change was to increase the diameter of the ball by about 10%. This had the effect of slowing the ball down and allowing more long rallies, thus making the game more attractive for spectators. It also made it easier to follow the fast-moving ball on television. Another change was to the scoring system. Instead of playing 5 sets (3 for ladies) up to 21 points, they now play 7 sets up to 11 points. I think it is intuitively 'obvious' that the second method tends to make the results more unpredictable and therefore the matches more interesting.
on Sep 8th, 2007, 6:36pm, SWF wrote:| You could continue with a similar analysis of probability of winning a set, and enough sets to win a match, but would need to have two probabilities depending on if a given player is serving or receiving. |
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P(GAME)
Let p = P(POINT), q = 1 - p
p4 + 4p4q + [10p4q2/(1 - 2pq)]
P(TIE-BREAK)
Let p = P(POINT), q = 1 - p
p7 + 7p7q + 28p7q2 + 84p7q3 + 210p7q4 + [462p7q5/(1 - 2pq)]
P(SET WITHOUT TIE-BREAK)
Let p = P(GAME), q = 1 - p
p6 + 6p6q + 21p6q2 + 56p6q3 + [126q6q4/(1 - 2pq)]
P(SET WITH TIE-BREAK)
Let p = P(GAME), q = 1 - p
p6 + 6p6q + 21p6q2 + 56p6q3 + 126q6q4 + 252p7q5 + 504p6q6*P(TIE-BREAK)
5-SET MATCH (MEN, NO TIE-BREAK IN 5th SET)
Let p = P(SET WITH TIE-BREAK), q = 1 - p
p3 + 3p3q + 6p2q2*P(SET WITHOUT TIE-BREAK)
3-SET MATCH (LADIES, ALWAYS TIE-BREAKS)
Let p = P(SET WITH TIE-BREAK), q = 1 - p
p2 + 2p2q
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| « Last Edit: Jan 19th, 2011, 6:08pm by ThudnBlunder » |
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srn437
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Re: Anyone For Tennis?
« Reply #29 on: Sep 8th, 2007, 7:49pm » |
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Doesn't that change the answer? Thunderblunder, quit blundering(no insult intended, just a pun).
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mikedagr8
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Re: Anyone For Tennis?
« Reply #30 on: Sep 8th, 2007, 7:51pm » |
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on Sep 8th, 2007, 7:49pm, srn347 wrote:| Doesn't that change the answer? Thunderblunder, quit blundering(no insult intended, just a pun). |
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Who is this 'Thunderblunder'? You need to stop making all this sh*t up. Seriously, learn to read.
Ok, I'll stop feeding him.
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| « Last Edit: Sep 8th, 2007, 7:52pm by mikedagr8 » |
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SWF
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Re: Anyone For Tennis?
« Reply #32 on: Sep 9th, 2007, 2:27pm » |
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Here is an attempt at comparing the game/set/match approach to a match to just playing until a certain number of points are scored. To keep it simple, ignored difference between serving and receiving probability and also ignored using the tie breakers for sets.
If probability of winning any given point is 0.6, using the formulas ThudanBlunder gives, probability of winning a game is 0.736, probability of winning a set is 0.966, and probability of winning a 5 set match is 0.9996. These values are higher than seen in practice, so some of the assumptions need improvement.
For a fair comparison to a match being a fixed number of points, need to find how many points are in a typical match. For 0.6 probability of winning each point, I come up with an average of 6.48 points per game, 8.07 games per set, and 3.10 sets per match, giving 162.4 points per match. If you make a match be over when the first player scores 98 points, the average number of points played to finish the match will be 162.9 points, assuming one player has 0.6 probability for each point. This results in probability of winning the match being 0.9976, which is less than with the game/set/match approach.
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srn437
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Re: Anyone For Tennis?
« Reply #33 on: Sep 9th, 2007, 3:36pm » |
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I had already said I understood it.
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #34 on: Sep 9th, 2007, 6:54pm » |
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on Sep 9th, 2007, 2:27pm, SWF wrote:If probability of winning any given point is 0.6, using the formulas ThudanBlunder gives, probability of winning a game is 0.736, probability of winning a set is 0.966, and probability of winning a 5 set match is 0.9996. These values are higher than seen in practice, so some of the assumptions need improvement.
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My formulae assume that the same person is always serving. So P(GAME) will be accurate, but not the others.
Similarly, 6.48 points per game ought to be accurate, but not the others.
Still, it is interesting that the first-to-x points probability appears to be less than the 'game, set, and match' one.
This comparison is valid, as we are assuming in each case that the service does not change hands.
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| « Last Edit: Feb 1st, 2009, 3:59am by ThudnBlunder » |
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srn437
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Re: Anyone For Tennis?
« Reply #35 on: Sep 9th, 2007, 7:05pm » |
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If the players alternate serving or let whoever scores a point serve, the probability would be?
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #36 on: Sep 9th, 2007, 7:10pm » |
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on Sep 9th, 2007, 7:05pm, srn347 wrote:| If the players alternate serving or let whoever scores a point serve, the probability would be? |
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f(p,q) of course. 
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| « Last Edit: Sep 9th, 2007, 7:11pm by ThudnBlunder » |
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srn437
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Re: Anyone For Tennis?
« Reply #37 on: Sep 9th, 2007, 7:13pm » |
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Which means?
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #38 on: Sep 9th, 2007, 7:15pm » |
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on Sep 9th, 2007, 7:13pm, srn347 wrote:
A function of p and q, of course.
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srn437
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Re: Anyone For Tennis?
« Reply #39 on: Sep 9th, 2007, 7:19pm » |
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That function is?
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #40 on: Sep 9th, 2007, 7:26pm » |
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on Sep 9th, 2007, 7:19pm, srn347 wrote:
A polynomial.
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srn437
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Re: Anyone For Tennis?
« Reply #41 on: Sep 9th, 2007, 7:30pm » |
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There are many different kinds of polynominals. Could you just say what it is specifically?
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #42 on: Sep 9th, 2007, 7:40pm » |
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on Sep 9th, 2007, 7:30pm, srn347 wrote:| Could you just say what it is specifically? |
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Well, I have worked out the first coefficient.
It is (-84/ )loge( ) 
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| « Last Edit: Sep 9th, 2007, 8:17pm by ThudnBlunder » |
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srn437
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Re: Anyone For Tennis?
« Reply #43 on: Sep 9th, 2007, 8:26pm » |
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Are you using a calculator of some sort?
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #44 on: Sep 9th, 2007, 8:30pm » |
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on Sep 9th, 2007, 8:26pm, srn347 wrote:| Are you using a calculator of some sort? |
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No, I can work out the above expression in my head. My brain is hard-wired for imaginary calculations.
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| « Last Edit: Feb 1st, 2009, 6:20am by ThudnBlunder » |
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srn437
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Re: Anyone For Tennis?
« Reply #45 on: Sep 9th, 2007, 8:32pm » |
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When you say he who knows speaks not and vice-verca, does that include typing?
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #46 on: Sep 9th, 2007, 9:05pm » |
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on Sep 9th, 2007, 8:32pm, srn347 wrote:| When you say he who knows speaks not and vice-verca, does that include typing? |
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Yes, he who types, speaks not.
He who speaks, types not.
It is written.
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| « Last Edit: Sep 10th, 2007, 3:52am by ThudnBlunder » |
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srn437
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Re: Anyone For Tennis?
« Reply #47 on: Sep 9th, 2007, 9:11pm » |
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I meant he who types knows not and he who knows types not. Is this also true?
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ThudnBlunder
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Re: Anyone For Tennis?
« Reply #48 on: Sep 9th, 2007, 9:14pm » |
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on Sep 9th, 2007, 9:11pm, srn347 wrote:| I meant he who types knows not and he who knows types not. Is this also true? |
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I knew what you meant, so typed not.
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| « Last Edit: Jun 13th, 2008, 7:33am by ThudnBlunder » |
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srn437
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Re: Anyone For Tennis?
« Reply #49 on: Sep 9th, 2007, 9:17pm » |
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Doesn't that mean that since we all type, we all speak not, and we all know?
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