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Probability and reality

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7plywood View Drop Down
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    Posted: 12/28/2012 at 11:34pm
Statistically when winning a game 10:9 you have 75% chance of actually winning it. Winning 10:8 gives you 87.5% probability of winning the game. Somehow that does not seem to match reality. I am pretty sure it does not. Could this be attributed to human nature = nervousness and unconscious relaxation, which causes lower concentration when you are on the winning side? I didn't actually perform any statistical analysis to prove that the theoretical probability does not match real results, so it might be just my impression...


Edited by 7plywood - 12/28/2012 at 11:57pm
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Post Options Post Options   Thanks (0) Thanks(0)   Quote DeIgado Quote  Post ReplyReply Direct Link To This Post Posted: 12/28/2012 at 11:42pm
The statistics are based off reality, so to answer your question, yes. You may not recognize the statistics in your wins because if you won, all that matters is the W. But when you lose after you have been ahead you tend to dwell on it. I would think that most people think more about their loses and how to improve rather than thinking about their wins.

This is what you should do, record all your games in which the score at some point in time is 10-8 or 10-9. Do this for 100 games and you should come up with a solid percentage for yourself. Get 100 other people to do this as well, and you will see that statistically, 85% and 75% are fairly correct.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote 7plywood Quote  Post ReplyReply Direct Link To This Post Posted: 12/28/2012 at 11:56pm
I am not talking about a narrow perspective of my own matches or any single player matches. I am rather considering high level players matches, making assumption of 50-50 chance of winning each point to calculate probability.
That should mathematically translate to actual statistics but somehow I don't think it does. Must be human factor involved, but again - I never actually recorded anything to calculate actual statistics.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote chu_bun Quote  Post ReplyReply Direct Link To This Post Posted: 12/28/2012 at 11:59pm
How do you get to these numbers?  They don't look right to me.  At 10:9 you are leading (barely) but not really winning.  Your opponent should be around your level, so your chance of winning the next point would be around 50%, maybe a little bit higher if you are on serve.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote 7plywood Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 12:11am
Simple probability.
Being 10:9 you have 50% chances of winning the next point, and if you don't win it - the game is tied up so it is 50% then.
1/2 + (1/2 * 1/2) = 3/4 = 75%
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Post Options Post Options   Thanks (0) Thanks(0)   Quote kenneyy88 Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 1:21am
It's not 50% because server has an advantage most of the time. 
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Post Options Post Options   Thanks (0) Thanks(0)   Quote 7plywood Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 1:28am
Originally posted by kenneyy88 kenneyy88 wrote:

It's not 50% because server has an advantage most of the time. 

But assuming that being 10:9 you are on serve with equal probability that you are not - it does not matter. It is still 50% that you win the point (human factor aside, as mentioned earlier).
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Post Options Post Options   Thanks (0) Thanks(0)   Quote 7plywood Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 1:33am
The real question is do you guys agree that in reality it seems that the guy who is losing 9:10 has higher chances of winning the game than just mere 25%?
If so, why is that?
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Post Options Post Options   Thanks (0) Thanks(0)   Quote gnome Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 4:11am
It does seem counterintuitive the player has a 25% chance of winning from 9:10.
Off the top of my head I would say between 35-45%.  The reason being is that the score looks even, and this should correspond to even chances - thats my "shortcut thinking".  I think the 25%  would be due to the feeling of "being on the edge of defeat".  Say the score is 5-10 and in one case the game is up to 21 and in the other it is 11.  I think in the 21 pt case it is quite common to win 4 or 5 pts in a row, but not so in the 11 case.

Of course, the guy about to win has the feeling of "please let me have this last point" too...
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Post Options Post Options   Thanks (0) Thanks(0)   Quote pingponger Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 4:39am
If every point proceeding from these scores were solely decided by the flip of the coin (which is never the case), then the probabilities initially mentioned are indeed correct.  In reality, there are other factors like the ones mentioned so far, which can pull the actual chances in either direction.

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Post Options Post Options   Thanks (0) Thanks(0)   Quote Speedplay Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 9:30am
Originally posted by pingponger pingponger wrote:

If every point proceeding from these scores were solely decided by the flip of the coin (which is never the case), then the probabilities initially mentioned are indeed correct.  In reality, there are other factors like the ones mentioned so far, which can pull the actual chances in either direction.


Agree. There are many ways to reach 10-9. What if the leading player is up 9-2, then the other player reaches 9-9 before the previously leading player wins another point to go to 10-9? In such a scenario, I would think the player who is down 9-10 would be the favourite to win the set.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote avova Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 10:26am
Phychological experiments demonstrate that humans  are incapable of "eyeing" probabilities. They almost always get probabilities waaay off.

If you feel 75% does not match the reality, then it is the way your brain distorts your perception of reality that makes you feel like that.

>> Somehow that does not seem to match reality. I am pretty sure it does
>> not.  I didn't actually perform any statistical analysis to prove that the
>> theoretical probability does not match real results
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Post Options Post Options   Thanks (0) Thanks(0)   Quote zeio Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 12:03pm
It should be that the probability of a player leading a game 10:9 has 50% chance of actually winning it(1 in 2 of winning the next point).  For the 10:8 scenario, the leading player still has 50% chance of winning whereas the other player has only 25% 33% chance of leveling the game to deuce(1 in 4 1 in 3 of winning the next two points), assuming the two events are statistically independent.  The key is there is a distinction between the percentage of winning and losing.

p.s. Corrected the percentage for the second scenario.  Forgot that the game ended at 11.  LOL

1       2       3
10:8  10:8  10:8
10:9  10:9  11:8
10:10 11:9


Edited by zeio - 12/29/2012 at 1:02pm
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Baal Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 12:04pm
Originally posted by 7plywood 7plywood wrote:

Statistically when winning a game 10:9 you have 75% chance of actually winning it. Winning 10:8 gives you 87.5% probability of winning the game. Somehow that does not seem to match reality. I am pretty sure it does not. Could this be attributed to human nature = nervousness and unconscious relaxation, which causes lower concentration when you are on the winning side? I didn't actually perform any statistical analysis to prove that the theoretical probability does not match real results, so it might be just my impression...


Impossible to test the hypothesis without actual data.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote chu_bun Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 12:24pm
I think this is not as simply as head you win, tail you lose.  There is a difference between winning a point and winning a set. 
The chance of winning a point at any given time is 50%, but the chance to win a set vary.  The calculations should change significantly when one or both of the scores reach 10.  Leading 1 point at 1-0 is not the same as 10-9. 
At 10-9 your chance of winning the next point and so a match is 50%.  But not winning does not mean you lose.  You cannot lose the set on *the next point alone* (0 chance).  In the case you lose, new rule win-by-2 rule is in effect.  But your chance of winning the set is still 50% (10-10 even score).  It's not the same as the chance of winning 2 consecutive points (1/2 * 1/2).
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Post Options Post Options   Thanks (0) Thanks(0)   Quote BH-Man Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 1:13pm
There are entirely to many variables unaccounted for in the OP's post. Serve, style, dominance of one player, playing level differneces, bad light on one side of court, you name it, too many variables.
I list one example below of a proven tournement winner's mindset.
 
A player who is stronger by at least a playing level when down 9-10 should be smiling to himself and saying "Now I got him right where I want him!" "He just lost the match and hasn't yet realized it!"
 
 
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Post Options Post Options   Thanks (0) Thanks(0)   Quote BH-Man Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 1:22pm
Last weekend I was at Sky TTC's tourney and faced one of the city's top players ( he is 2 levels better than me, he plays div 0 (ZERO !!!) in our city and is easily a 2350-2400ish player) in the round of 16. It was 2-2 and I was down 8-10 and HE was serving. I ended up fighting and winning points to deuce, then 12-10. In the round of 8, I faced a div 1 city player and also found my self down 0-2 in sets down 6-10 in points game 3. I fought, won game 3 at deuce, won game 4, and was down 8-10 at 2-2. I fought, made rallies, got to deuce, and won 12-10.
 
What were the odds of coming back down 8-10 vs clearly superior players TWICE in back to back matches?
 
We could try to calculate it, but it simply wouldn't apply on the court where players fight and win or lose in changing emotions, confidence and other dynamics not expressed here. Each point is and in this case, was, a new start with odds not the deciding factor, but other things decided those two matches.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote pingponger Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 3:23pm
+10 for the concept of 'momentum' which was effectively described above.  To illustrate it, a race car in pole position is closer to the finish line, but the second car might be going faster (fast enough to pass and take the checkered flag) .
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Baal Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 4:06pm
What is actually needed but not available (it seems) is what the actual percentages (with large and broad sample size) of games won when a player is leading 10-9.  Without that, it is impossible to determine if there is statistically significant deviation from what would be expected.  Of course, how to calculate the expected result could be debated.  Perhaps easiest way is to assume that at 10-9, both players have 50% chance of winning all subsequent points.  That assumes players are equal playing level, which is certainly not always true. 

On the other hand, there is a complication in collecting real numbers, which is how to correct for the fact that a lot of the time the guy who is ahead 10-9 is just better.  Thus, consider a match in which the current game is 10-9 but scores of previous games are 11-4, 11-5, 11-5.  You would want to not treat that 10-9 game the same way as a 10-9 game in a match where previous games were 10-12, 11-9, 9-11, 13-11. 

None of this is as easy as it sounds. 
 
     

Edited by Baal - 12/29/2012 at 4:07pm
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Anton Chigurh Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 4:31pm
Originally posted by Baal Baal wrote:

What is actually needed but not available (it seems) is what the actual percentages (with large and broad sample size) of games won when a player is leading 10-9.  Without that, it is impossible to determine if there is statistically significant deviation from what would be expected.  Of course, how to calculate the expected result could be debated.  Perhaps easiest way is to assume that at 10-9, both players have 50% chance of winning all subsequent points.  That assumes players are equal playing level, which is certainly not always true.  

I'm admittedly (and embarrassingly) a little rusty on my stats right now since I haven't used them in a bit, but I'm fairly certain that what you mentioned is exactly how the analysis would be performed: Collect the actual data and see if the actual data significantly differs from the assumption of 50%. If so, this would call the basic assumptions of the original post into question. My intuition is that yes, the data would often significantly differ from 50%... though, again, that itself may vary as a function of the players playing each other at the time. 

There is a ~1900 player at my club who I occasionally get games off of... but he definitely wins 85%-90% if the time. If were up 10 to 9, I do not think it's a safe assumption that there is a 50/50 chance either of us would win the subsequent point. I think the same could be said for various high level pros. Their games are admittedly more stable, etc., but there is still some meaningful variability there that tips the scales in one way or another in most match ups. 

Originally posted by Baal Baal wrote:


On the other hand, there is a complication in collecting real numbers, which is how to correct for the fact that a lot of the time the guy who is ahead 10-9 is just better.  Thus, consider a match in which the current game is 10-9 but scores of previous games are 11-4, 11-5, 11-5.  You would want to not treat that 10-9 game the same way as a 10-9 game in a match where previous games were 10-12, 11-9, 9-11, 13-11. 

None of this is as easy as it sounds. 

The main problem I foresee is that the foundational assumption of most statistical analyses is the assumption of the random distribution of error. That is, it is assumed that each observation is independent of the events prior to it. I do not think that is the case in this situation, regarding the score in a game.

I am confident that some kind of procedure could be used to provide some useful estimations, but as you mentioned Baal, this would not be nearly as easy/straight forward as it one might think. 



Edited by Anton Chigurh - 12/29/2012 at 9:59pm
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Post Options Post Options   Thanks (0) Thanks(0)   Quote 7plywood Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 6:45pm
Seems like almost everybody directly or indirectly agrees that probability of actually winning a game after being down 9:10 "feels" higher than just mathematical 25%.

Actually - using just robotic probability and assuming only what we know: the score being 10:9 - we can conclude that the player who has 10 is more likely to be better by that factor (10 to 9) from the player who has 9 points, so going by that the probability of the player having 9 points coming back should be actually below 25%.

Unfortunately I don't have time to perform statistical analysis of (for example) past ITTF tournaments, but it would be an interesting study about "choking" in Table Tennis.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Baal Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 9:56pm
Actually, I'm not sure where the 25% number comes from.  Maybe my brain has just shut off today.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote chu_bun Quote  Post ReplyReply Direct Link To This Post Posted: 12/29/2012 at 11:28pm
25% seems to be too low.  Why dont everybody records the result of 9-10 matches that they have for a week or so and post it here.  Hopefully the sample is big enough to come up with something meaningful.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote infinite_loop Quote  Post ReplyReply Direct Link To This Post Posted: 01/03/2013 at 5:37am
I agree with avova and Delgado.

People are known to be bad at estimating such probabilities. We have a ton of cognitive biases that lead us astray. In this case, it is "availability error" of overestimating the frequency of unusual events and underestimating ordinary ones.

Remember that match where your opponent had a match point, leading 3 games to 2 and 10-9 in 6th game, and you fought back to take the game, and eventually the match? That registers in your mind more strongly compared to an ordinary win from 10-9 to 11-9.

Trust your math more than your perception. your math is correct assuming ideal model (equal skill players). I think reality will be close to 75-25 if you verify with a large unbiased sample. Psychological factors of leading and trailing player will mostly even out.

In practice, it will be just a little different from 75-25 due to these complications to calculation, as discussed in previous posts
1. Stronger player would more frequently lead 10-9 than trail 9-10, so it increases his chance a little
2. In a hypothetical scenario with absolutely equal players, the player leading 10-9 is a little more likely to be receiving than serving, so it decreases his chance a little
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