Weak strong era - Interesting probability experiment (please participate)

[kragster]

The discussion around weak and strong era really should first focus on - is it statistically possible for one era of players to be weaker or stronger than another?

Would love to carry out a simple experiment to prove/disprove a point. Please go to the site below and do this coin flipping experiment

http://www.mathsonline.co.uk/nonmembers/resource/prob/coins.html


1. Select 10 coins and 1000 trials.
2. Report back to me the number of times you got 5Hs in a row, 9 Heads in a row and 10 heads in a row
My numbers are

5 Heads:256
9 Heads:5
10 Heads:1

Once we have enough results , I would like to draw the following conclusions:

1) Let's assume the top 100 tennis players of any era have a skill/ability level (includes physical/mental skill) from 1 to 10.

2) The average competitiveness of an era is almost always the same i.e in the coin tossing experiment we SHOULD see that most folks have approximately the same number of times when they got 5 Heads. One piece of evidence to prove this is by looking into the avg ATP points for the 50th ranked player in the world every year. If you go back and look at the results you will find with remarkable consistency that these players have between 850-950 ATP ranking points. I can predict with an extremely high level of confidence that the 50th ranked player next year will have between 850-950 pts.

2) Like any distribution though, once you get to the edges, there is a lot more uncertainty. The number of times folks got 9 Heads or 10 Heads will be quite different. The tennis version of this is that it is much harder to predict the avg ATP points of the # 1 ranked player .

3) An era where a player racks up a bunch of slams could be a function of that person being a 10 and the next best players being 9's. An era where players split slams could be where the top players are all 10's or all 9's. In reality it is hard to tell which is the case. Is a competitive era because we have more than one 10 , or because we have multiple 9s. However statistically it would be much less likely that the reason an era is competitive is because it has a bunch of 10s . The coin tossing experiment should show very few people with more than one 10.


4) One benchmark we have that should be relatively stable is the one we established in point (1) that does not shift across time. And that benchmark is the average level of the era. For the most part (discounting advances in technology/health etc) the average level of an era is constant. So a top player's winning % vs the average crowd (lets say players ranked 40-60) should tell you how strong that player is. That could be one way to tell if the player is a 9 or a 10.


Of course there are numerous realities not reflected in the statistics. Player's levels change , they could be injured , the 'ability' of a player is not an absolute quantity that is independent of surface conditions. But by and large, once you account for major changes (such as Australian open becoming a slam) I think it is fair to make comparisons across eras.

 

[kragster]

C'mon folks, be a sport!

 

[LuckyR]

What is your premise? And while we are at it, how does one know if an era is "strong". What would a strong era look like, statistically?

 

[mightyrick]

What is your premise? And while we are at it, how does one know if an era is "strong". What would a strong era look like, statistically?

I don't know what the mathematical model would be for tennis, but for baseball... the layman's description is that a weak era would be a period of years where there are only a very few teams that beat the hell out of each other (evenly matched), but these teams absolutely crush the rest of the teams.

In tennis, this probably translates to the following: Over a period of years, if you have a #1 and #2 with a competitive H2H... but they absolutely crush the rest of the rankings... then this would be a weak era. It isn't just a balanced H2H of the top competitors that counts. It is also how badly they beat the rest of the field.

I think tennis would be easier to model. You probably could just use margins of ATP points within the top 10 over a period of years as a prime indicator of strength of era.

 

[LuckyR]

I don't know what the mathematical model would be for tennis, but for baseball... the layman's description is that a weak era would be a period of years where there are only a very few teams that beat the hell out of each other (evenly matched), but these teams absolutely crush the rest of the teams.

In tennis, this probably translates to the following: Over a period of years, if you have a #1 and #2 with a competitive H2H... but they absolutely crush the rest of the rankings... then this would be a weak era. It isn't just a balanced H2H of the top competitors that counts. It is also how badly they beat the rest of the field.

I think tennis would be easier to model. You probably could just use margins of ATP points within the top 10 over a period of years as a prime indicator of strength of era.

OK, so what would a strong era look like?

 

[kragster]

I don't know what the mathematical model would be for tennis, but for baseball... the layman's description is that a weak era would be a period of years where there are only a very few teams that beat the hell out of each other (evenly matched), but these teams absolutely crush the rest of the teams.

In tennis, this probably translates to the following: Over a period of years, if you have a #1 and #2 with a competitive H2H... but they absolutely crush the rest of the rankings... then this would be a weak era. It isn't just a balanced H2H of the top competitors that counts. It is also how badly they beat the rest of the field.

I think tennis would be easier to model. You probably could just use margins of ATP points within the top 10 over a period of years as a prime indicator of strength of era.

I think what you are defining is a competitive vs non-competitive era. This is different from weak vs strong era. The definition of a weak era refers to an era where the average skill-set is weak THUS leading to a few teams dominating.

In theory you could have
1) A non competitive weak era (bunch of poor teams with a few superstars)
2) A competitive weak era ( bunch of poor teams with no superstars)


What I was trying to prove is that there is no such thing as a weak era. But you could have a more competitive vs less competitive era based on how many outliers you have.

 

[NatF]

For the sake of the experiement. I had;

5: 262
9: 11
10: 2

 

[qwertre]

246 fives
11 nines
0 tens

 

[90's Clay]

Obviously certain eras produce more talent then others. There are weaker and stronger eras because of this. You don't get the same talent through every era. Talent, threats, competition is constantly changing

One era you could have a solid top 10 full of highly talented players all threats to the top. Other eras could you have a solid top 2-3 guys, while the rest not even worthy of calling themselves top 5-10 in the world.(Like this era for instance.. I mean Ferrer a top 4? Really? Would he be top 4 in the 80s or early 90s for instance?)

Senseless empty numbers and stats can't explain talent, threats and competition. It only tells "some" of the story. Not the entire story

 

[Povl Carstensen]

Talent, threats, competition is constantly challenging.

True, it is constantly challenging.

 

[Povl Carstensen]

It is also how badly they beat the rest of the field.

But that is relative...

 

[Povl Carstensen]

What is your premise? And while we are at it, how does one know if an era is "strong". What would a strong era look like, statistically?

I think this is basically the premise:

3) An era where a player racks up a bunch of slams could be a function of that person being a 10 and the next best players being 9's. An era where players split slams could be where the top players are all 10's or all 9's. In reality it is hard to tell which is the case. Is a competitive era because we have more than one 10 , or because we have multiple 9s. However statistically it would be much less likely that the reason an era is competitive is because it has a bunch of 10s . The coin tossing experiment should show very few people with more than one 10.[/B]

 

[mightyrick]

I think what you are defining is a competitive vs non-competitive era. This is different from weak vs strong era. The definition of a weak era refers to an era where the average skill-set is weak THUS leading to a few teams dominating.

No. I'm saying that a strong era is an era where there are several top players who are evenly matched with one another but also don't dominate the field in a lopsided way. A weak era is where there are only a couple of top players and they absolutely crush the field in a lopsided way.

You are making it subjective by bringing in this "average skillset" thing. Stick only with facts.

You have to start with the axiom that for an era, a period of years, the best players available at that time are playing the game professionally.

For a given era, there will be a best player and second best player and third best player... and so on. The strength of the era is defined by how competitive those best players are with one another... and how badly those players compete with the rest of the field.

I'm not talking about comparing eras to one another. That is impossible. However, what you can compare is one player's relative accomplishment in their era to another player's accomplishment in another era.

That is how the GOATs are derived.

 

[Povl Carstensen]

Is Federer a 10 or a lucky 9...

 

[LuckyR]

No. I'm saying that a strong era is an era where there are several top players who are evenly matched with one another but also don't dominate the field in a lopsided way. A weak era is where there are only a couple of top players and they absolutely crush the field in a lopsided way.

You are making it subjective by bringing in this "average skillset" thing. Stick only with facts.

You have to start with the axiom that for an era, a period of years, the best players available at that time are playing the game professionally.

For a given era, there will be a best player and second best player and third best player... and so on. The strength of the era is defined by how competitive those best players are with one another... and how badly those players compete with the rest of the field.

I'm not talking about comparing eras to one another. That is impossible. However, what you can compare is one player's relative accomplishment in their era to another player's accomplishment in another era.

That is how the GOATs are derived.

Your description, which I believe reflects a common understanding of the terminology, is IMO flawed, hence my belief that there is no such thing as "weak" and "strong" eras, since whatever criteria you use to define them can be derived from their opposite.

For example you call Strong "an era where there are several top players who are evenly matched with one another but also don't dominate the field in a lopsided way". Well what happens statistically if you take such an era and add the GOAT, suddenly you get: "a couple of top players (or just one) and they absolutely crush the field in a lopsided way", which is your definition of Weak.

Adding the GOAT turns Strong to Weak? Huh? It should make the era Stronger, not Weaker.

 

[Jules]

5: 278
9: 10
10: 0

 

[djones]

Where does 1 era start or end?
"Era" is relative.

 

[forzamilan90]

5: 221
9: 9
10: 0

 

[monfed]

Oh look the attention seeker strikes again with another useless thread.

 

[FEDERERNADAL13]

5: 249
9: 13
10: 0

 

[tvizz]

I think the sim is flawed in some way, we should have half the number of 10s as 9s but instead we see more of a 10/1 split.

 

[mightyrick]

For example you call Strong "an era where there are several top players who are evenly matched with one another but also don't dominate the field in a lopsided way". Well what happens statistically if you take such an era and add the GOAT, suddenly you get: "a couple of top players (or just one) and they absolutely crush the field in a lopsided way", which is your definition of Weak.

I'm saying that in a strong era, a GOAT would not completely dominate. But in a weak era, they would absolutely dominate.

I mean break this down to something very simple to understand what I'm talking about. Let's say there are only three people in the ATP. These three people are evenly matched. They beat the hell out of each other... they switch spots in the rankings frequently.

Then you add the GOAT. The GOAT wins 85% of their matches against the other three. Now, one of the other three retires. A new player comes in. The GOAT now wins 88% of their matches against the other three.

So what happened? The field got weaker.

Okay, so let's continue this line. Another one of those other three retires. The new replacement player that comes in is better than the others... but still not as good as the GOAT. The GOAT now wins 78% of their matches against the other three.

So what happened? The field got stronger. The GOAT wins less, but he's still the GOAT.

Win percentage alone is not enough to determine a GOAT. Nor is strength of the field alone. You have to factor how the rest of the field does against the GOAT as well as how the rest of the field performs against itself.

But having Federer, Nadal, and Djokovic together in the same era does not make it weak. It makes it strong. And in this configuration, the best player is not going to win as much.

This is why I contend that Graf is the GOAT. Her era had very strong players who all beat the hell out of each other and traded spots continually. They even beat Graf many times. But over the course of her era and career, Graf dominated all of them -- leading to over 20 slams. But as great as that accomplishment was, it certainly wasn't lopsided. She took a lot of lumps. Strong era.

 

[Feather]

Oh look the attention seeker strikes again with another useless thread.

Among the Rafa fans who try to convince everyone that Fed dominated weak era, among the Federer fans who try to impose on everyone that Roger is the GOAT and among the Djokovic fans who think there was no Tennis beyond 2008, guys like kragster is a revelation. At least he comes up with some light threads which at least few of us consider good

 

[Flash O'Groove]

Very interesting thread. You ask a very important era: What are the probabilities that an era is weaker or stronger than another?

The 2000-2007 era is considered by many as weak era, while 2008-2012 is considered by many as a strong era, the golden era of tennis.

Two hypothesis:

1) The strength of the era is only the result of a subjective judgment: if a player crush his rivals, it is a weak era, or a strong era, depending of the subjectivity of the viewer.

2) The strength or weakness of the era is objective. In this case, what are the reason which make one era a weak or a strong one? Randomness? What are the probability that in a definite time span, the tennis world produce a different number of great player, or with a different level, than in another time-span? Your experiment will show that the probability is very small.

--> is it only a question of probability, or some "outside" matter affect the production of great player? In this case which one? What are the variables we can use to explain that the 2004-2007 era is a weak one, and what are the variables we can use to explain that 2008-2012 era is a strong one?

Result of the coins toss:
5: 248
9: 10
1: 0

 

[Flash O'Groove]

Among the Rafa fans who try to convince everyone that Fed dominated weak era, among the Federer fans who try to impose on everyone that Roger is the GOAT and among the Djokovic fans who think there was no Tennis beyond 2008, guys like kragster is a revelation. At least he comes up with some light threads which at least few of us consider good

Whatever Kragster aims to show (I don't know if is more a Fed or someone else fan), he made effort to build a constructive experiment, which might lead to a better understanding of tennis. Some people may be afraid to find something which go against their beliefs...

 

[FedError]

I think the sim is flawed in some way, we should have half the number of 10s as 9s but instead we see more of a 10/1 split.

The numbers are correct. There is only 1 way you can land all heads, and that is to get 10 heads in a row.

But there are 10 different ways you can land 9 heads, each of these with equal chance as landing 10 heads:

Thhhhhhhhh
hThhhhhhhh
hhThhhhhhh
hhhThhhhhh
hhhhThhhhh
hhhhhThhhh
hhhhhhThhh
hhhhhhhThh
hhhhhhhhTh
hhhhhhhhhT

So that explains the 10/1 split.

 

[FedError]

Also here are the averages you should see if you took an infinite number of trials:

0 heads: 0.9765625
1 head: 9.765625
2 heads: 43.9453125
3 heads: 117.1875
4 heads: 205.078125
5 heads: 246.09375
6 heads: 205.078125
7 heads: 117.1875
8 heads: 43.9453125
9 heads: 9.765625
10 heads: 0.9765625

---

On another note, if you wanted the probabilities on how many 10s you would see in a single trial, here they are.

zero 10s: 0.3764
one 10: 0.3680
two 10s: 0.1797
three 10s: 0.0584
four 10s: 0.0142
five 10s: 0.0028
six 10s: 0.0004
more than six 10s: approx 0.0001

 

[moonballs]

I'm saying that in a strong era, a GOAT would not completely dominate. But in a weak era, they would absolutely dominate.

I mean break this down to something very simple to understand what I'm talking about. Let's say there are only three people in the ATP. These three people are evenly matched. They beat the hell out of each other... they switch spots in the rankings frequently.

Then you add the GOAT. The GOAT wins 85% of their matches against the other three. Now, one of the other three retires. A new player comes in. The GOAT now wins 88% of their matches against the other three.

So what happened? The field got weaker.

Okay, so let's continue this line. Another one of those other three retires. The new replacement player that comes in is better than the others... but still not as good as the GOAT. The GOAT now wins 78% of their matches against the other three.

So what happened? The field got stronger. The GOAT wins less, but he's still the GOAT.

Win percentage alone is not enough to determine a GOAT. Nor is strength of the field alone. You have to factor how the rest of the field does against the GOAT as well as how the rest of the field performs against itself.

But having Federer, Nadal, and Djokovic together in the same era does not make it weak. It makes it strong. And in this configuration, the best player is not going to win as much.

This is why I contend that Graf is the GOAT. Her era had very strong players who all beat the hell out of each other and traded spots continually. They even beat Graf many times. But over the course of her era and career, Graf dominated all of them -- leading to over 20 slams. But as great as that accomplishment was, it certainly wasn't lopsided. She took a lot of lumps. Strong era.

This line of thinking is exactly OP's simple math example is trying to disapprove. You can see from the large variation of the 9s and 10s among different trials using exactly SAME coins (meaning same field) it is very hard to tell whether the degree of competitiveness among the top two is true signal or randon occurance.

 

[THUNDERVOLLEY]

Oh look the attention seeker strikes again with another useless thread.

...which you served to him on a platter by posting.

 

[mightyrick]

Among the Rafa fans who try to convince everyone that Fed dominated weak era, among the Federer fans who try to impose on everyone that Roger is the GOAT and among the Djokovic fans who think there was no Tennis beyond 2008, guys like kragster is a revelation. At least he comes up with some light threads which at least few of us consider good

Yeah, I'm not a fan of the math experiment, but I know that all I've been trying to say is that it is possible for someone to be the GOAT in a weak era -- just as possible as it is for someone to be the GOAT in a strong era.

The criteria that most tennis people make for a GOAT is very self-serving. As someone else pointed out, it usually is some over-zealous fan. Their criteria is usually something ridiculously simple like:

* The GOAT wins most slams.

- or -

* The GOAT consistently beats the guy who wins the most slams.

Yet neither one is correct. Both are completely flawed and take nothing in context.

 

[kragster]

Also here are the averages you should see if you took an infinite number of trials:

0 heads: 0.9765625
1 head: 9.765625
2 heads: 43.9453125
3 heads: 117.1875
4 heads: 205.078125
5 heads: 246.09375
6 heads: 205.078125
7 heads: 117.1875
8 heads: 43.9453125
9 heads: 9.765625
10 heads: 0.9765625

---

Awesome, this is perfect to compare the random samples from folks with averages!

 

[kragster]

So I think we have enough samples from folks to show a trend. Thanks also to ******** for providing the statistical average as comparison:

# 5 Heads folks got : 256, 262, 246,278,221,249,248 ( Statistical average =246)
# 9 heads folks got : 5,11,11,10,9,13,10 (Statistical average =10)

# 10 Heads folks got: 1,2,0,0,0,0,0 (Statistical avg = 1)

The point I am trying to make here is that pretty much until you get to the extreme skill levels (the 10), you are going to see remarkable consistency. Every era should (ON AVERAGE) have the same number of very good players. However, when it comes to the best players, some eras may have zero 10s and some eras may have two 10s (Of course it would be very unlikely for an era to have four or five 10s)

In reality , skill levels would not be so discrete, there could be 9.5's . But the point stands that it would be a statistical aberration to have an era where you don't have a number of 'very good' players i.e. the 9's. Which means that when you do see 1 player dominate, it is a much higher chance that the player is a 10 and the next best folks are 9, than the chance that the player is a 9 but everyone else is an 8.

 

[tvizz]

Thank you for clarifying the odds, that helps.

I was thinking of the odds of flipping 9 heads and then the odds of the next coin.

 

[LuckyR]

I'm saying that in a strong era, a GOAT would not completely dominate. But in a weak era, they would absolutely dominate.

I mean break this down to something very simple to understand what I'm talking about. Let's say there are only three people in the ATP. These three people are evenly matched. They beat the hell out of each other... they switch spots in the rankings frequently.

Then you add the GOAT. The GOAT wins 85% of their matches against the other three. Now, one of the other three retires. A new player comes in. The GOAT now wins 88% of their matches against the other three.

So what happened? The field got weaker.

Okay, so let's continue this line. Another one of those other three retires. The new replacement player that comes in is better than the others... but still not as good as the GOAT. The GOAT now wins 78% of their matches against the other three.

So what happened? The field got stronger. The GOAT wins less, but he's still the GOAT.

Win percentage alone is not enough to determine a GOAT. Nor is strength of the field alone. You have to factor how the rest of the field does against the GOAT as well as how the rest of the field performs against itself.

But having Federer, Nadal, and Djokovic together in the same era does not make it weak. It makes it strong. And in this configuration, the best player is not going to win as much.

This is why I contend that Graf is the GOAT. Her era had very strong players who all beat the hell out of each other and traded spots continually. They even beat Graf many times. But over the course of her era and career, Graf dominated all of them -- leading to over 20 slams. But as great as that accomplishment was, it certainly wasn't lopsided. She took a lot of lumps. Strong era.

Others have made this observation but I'll resurect it here: your post is freely substituting 'strong' with 'competitive'. No doubt, W/L percentages (as you and many others commonly use) will reflect competitiveness (or lack thereof) quite accurately. But you could use the same algorhythm on a High School's results over the years and no one would think that would reflect strength of play, since the stats would be similar in SoCal beach communities (who are extremely strong) and midwe5t rural communities where teams would be relatively speaking much weaker.

You could put the difference between "competitive" and "strong" as just a semantic slight of hand, but IMO they address completely seperate issues.

 

[mightyrick]

In reality , skill levels would not be so discrete, there could be 9.5's . But the point stands that it would be a statistical aberration to have an era where you don't have a number of 'very good' players i.e. the 9's. Which means that when you do see 1 player dominate, it is a much higher chance that the player is a 10 and the next best folks are 9, than the chance that the player is a 9 but everyone else is an 8.

I'm sorry, but the bolded statement is bad science and is a complete non sequitur. What is this? Dungeons and Dragons? You're going to roll ten 1d2 dice as a model for the distribution of "skill level" within a tennis field? You don't need to make it random outcomes. The ATP already has a bunch of history and uses a pretty good point system. It is simple to look at distributions of positive/negative results. You don't have to make it up with dumb coin flips.

Here is the top-10 from the end of 2004 to end of 2009. It shows the percentage of ATP points held by each player in the Top-10 -- relative to the Total Points held by the Top-100. It also shows the percentage of ATP points held in total by the Top-10... relative to the Total Points held by the Top-100.

Ok, so what can we learn from this?

In 2004, the Top-10 held 30% of the ATP points in the Top-100. In 2009, the Top-10 held 41% of the ATP points in the Top-100. So which Top-10 do you think is stronger or weaker based on that? You'd probably say 2009 was stronger.

Now you look at the percentage of point distributions of the Top-10 within those years. In 2004, it was 6/4/4/3/3/2/2/2/2/2. In 2009, it was 7/6/6/5/5/3/3/2/2/2. So which Top-10 do you think is stronger or weaker based on that? You'd probably say 2009 was stronger.

I could have extended this to the Top-250 and the results would be the same.

Now, let's look at some lopsidedness. In 2006, the percentage of point distributions in the Top-10 was 8/4/3/2/2/2/2/2/2/2. Much more lopsided than in 2004 or 2009. Was the field better in 2006 or worse? Were the players in the Top-10 itself more or less competitive with the field and each other in 2006?

If anyone wants to take a look at it, they can. Just run the numbers and start slicing them. But don't make things up or try to serve your own conclusions with bad science.

 

[darrinbaker00]

The discussion around weak and strong era really should first focus on - is it statistically possible for one era of players to be weaker or stronger than another?

Would love to carry out a simple experiment to prove/disprove a point. Please go to the site below and do this coin flipping experiment

http://www.mathsonline.co.uk/nonmembers/resource/prob/coins.html


1. Select 10 coins and 1000 trials.
2. Report back to me the number of times you got 5Hs in a row, 9 Heads in a row and 10 heads in a row
My numbers are

5 Heads:256
9 Heads:5
10 Heads:1

Once we have enough results , I would like to draw the following conclusions:

1) Let's assume the top 100 tennis players of any era have a skill/ability level (includes physical/mental skill) from 1 to 10.

2) The average competitiveness of an era is almost always the same i.e in the coin tossing experiment we SHOULD see that most folks have approximately the same number of times when they got 5 Heads. One piece of evidence to prove this is by looking into the avg ATP points for the 50th ranked player in the world every year. If you go back and look at the results you will find with remarkable consistency that these players have between 850-950 ATP ranking points. I can predict with an extremely high level of confidence that the 50th ranked player next year will have between 850-950 pts.

2) Like any distribution though, once you get to the edges, there is a lot more uncertainty. The number of times folks got 9 Heads or 10 Heads will be quite different. The tennis version of this is that it is much harder to predict the avg ATP points of the # 1 ranked player .

3) An era where a player racks up a bunch of slams could be a function of that person being a 10 and the next best players being 9's. An era where players split slams could be where the top players are all 10's or all 9's. In reality it is hard to tell which is the case. Is a competitive era because we have more than one 10 , or because we have multiple 9s. However statistically it would be much less likely that the reason an era is competitive is because it has a bunch of 10s . The coin tossing experiment should show very few people with more than one 10.


4) One benchmark we have that should be relatively stable is the one we established in point (1) that does not shift across time. And that benchmark is the average level of the era. For the most part (discounting advances in technology/health etc) the average level of an era is constant. So a top player's winning % vs the average crowd (lets say players ranked 40-60) should tell you how strong that player is. That could be one way to tell if the player is a 9 or a 10.


Of course there are numerous realities not reflected in the statistics. Player's levels change , they could be injured , the 'ability' of a player is not an absolute quantity that is independent of surface conditions. But by and large, once you account for major changes (such as Australian open becoming a slam) I think it is fair to make comparisons across eras.

For those of you suffering from insomnia, just bookmark this post.

 

[Povl Carstensen]

In reality , skill levels would not be so discrete, there could be 9.5's . But the point stands that it would be a statistical aberration to have an era where you don't have a number of 'very good' players i.e. the 9's. Which means that when you do see 1 player dominate, it is a much higher chance that the player is a 10 and the next best folks are 9, than the chance that the player is a 9 but everyone else is an 8.

That is pretty much it.

 

[tennis_pro]

Ok, so what can we learn from this?

In 2004, the Top-10 held 30% of the ATP points in the Top-100. In 2009, the Top-10 held 41% of the ATP points in the Top-100. So which Top-10 do you think is stronger or weaker based on that? You'd probably say 2009 was stronger.

It could just as easily mean that guys ranked 10th and lower in 2009 were worse than the players ranked lower than 10 in 2004 so they gathered even less points.

People tend to agree that players out of the top 10 are non-factors.

 

[Povl Carstensen]

It could just as easily mean that guys ranked 10th and lower in 2009 were worse than the players ranked lower than 10 in 2004 so they gathered even less points.

People tend to agree that players out of the top 10 are non-factors.

Deleted.
10chx.

 

[FedError]

If you'd like to generate results on a whim (without having to wait for the long simulation), try this code; it talkes less than 1 second to run the trial.


var a = new Array(0,0,0,0,0,0,0,0,0,0,0), n;
for (var i = 0; i < 1000; i++){
n = 0;
for (var j = 0; j < 10; j++){
n += (Math.random() < 0.5);
}
a[n]++;
}
for (var i = 0; i < 11; i++){
console.log(i + ' heads: ' + a);
}


You can paste it in an online javascript console like http://repl.it/languages/JavaScript. Just paste the code on the left hand side and press the arrow to execute it.

Edit: Acually jsconsole.com is a better site since it doesn't start giving the same results all the time. Just paste it at the top and press enter.

Edit2: Probably the easiest way is to press Ctrl+Shift+K if you're using Firefox or Ctrl+Shift+J if you're using Chrome and paste the code there.