I have a real-life Probability problem

[Wi_N]

TL;DR Summary

Regarding match fixing in a wrestling tournament.

I'll try to keep this short. Kazakhstan just hosted the world championship wrestling tournament and I noticed that they did exceptionally well this time around, hometown psychological advantage aside they finished in 2nd place. Last year they finished 13th in Budapest. So immediately I knew something was wrong. And i figured it out. In the brackets draw for the 10 weight classes, they drew 8 straight brackets with no Russian wrestler in them (Russian wrestlers are the best). The last 2 weight classes 97kg and 130kg wouldn't have made a difference for them if a Russian was there or not, I won't go into why that is.

So some simple math, heads or tails i drew the conclusion that the odds of Kazakhstan drawing a bracket TOP or BOTTOM with no Russian in them has to be 1 in 128. Now this is heavily simplified I'm sure as some brackets had more competitors in them. But usually a weight class has about 32-24 wrestlers in them, in an elimination tournament setup. What are the odds of 2 countries not drawing the same bracket 8 times in a row? Please remember there are only 2 Brackets TOP and Bottom.

 

[PeroK]

Summary: Regarding match fixing in a wrestling tournament.

I'll try to keep this short. Kazakhstan just hosted the world championship wrestling tournament and I noticed that they did exceptionally well this time around, hometown psychological advantage aside they finished in 2nd place. Last year they finished 13th in Budapest. So immediately I knew something was wrong. And i figured it out. In the brackets draw for the 10 weight classes, they drew 8 straight brackets with no Russian wrestler in them (Russian wrestlers are the best). The last 2 weight classes 97kg and 130kg wouldn't have made a difference for them if a Russian was there or not, I won't go into why that is.

So some simple math, heads or tails i drew the conclusion that the odds of Kazakhstan drawing a bracket TOP or BOTTOM with no Russian in them has to be 1 in 128. Now this is heavily simplified I'm sure as some brackets had more competitors in them. But usually a weight class has about 32-24 wrestlers in them, in an elimination tournament setup. What are the odds of 2 countries not drawing the same bracket 8 times in a row? Please remember there are only 2 Brackets TOP and Bottom.

If you have 10 weight categories, then the odds of Kazakhstan being in the opposite bracket to Russia in 8/10 categories (or more) is 7/128. That's quite low.

This assumes a straight draw without seeding, and one Russian and one Kazakhstani in each category. You need to be careful that there aren't legitimate reasons why the wrestlers got drawn apart: like they were the top two seeds.

You also need to be careful about excluding data for various reasons. It's better to treat this as 8/10 and not treat this as 8/8 with two categories excluded for whatever reason. You can potentially always find reasons to exclude data that contradicts a hypothesis.

 

[Wi_N]

If you have 10 weight categories, then the odds of Kazakhstan being in the opposite bracket to Russia in 8/10 categories (or more) is 7/128. That's quite low.

This assumes a straight draw without seeding, and one Russian and one Kazakhstani in each category. You need to be careful that there aren't legitimate reasons why the wrestlers got drawn apart: like they were the top two seeds.

You also need to be careful about excluding data for various reasons. It's better to treat this as 8/10 and not treat this as 8/8 with two categories excluded for whatever reason. You can potentially always find reasons to exclude data that contradicts a hypothesis.

do the number of wrestlers in each weight class and unevenly spread between the two brackets play a role at all? also kazakhstan had no top seeds at all. is it safe to safe some manipulation of the draws occurred?

 

[PeroK]

do the number of wrestlers in each weight class and unevenly spread between the two brackets play a role at all?

Not really. There's a 50-50 chance of any two wrestlers being in the same bracket and a 50-50 chance of being in the opposite bracket. That's the fundamental assumption. If there are more wrestlers in one bracket than the other, then you'd have to explain in more detail how the draw is made.

 

[Wi_N]

Not really. There's a 50-50 chance of any two wrestlers being in the same bracket and a 50-50 chance of being in the opposite bracket. That's the fundamental assumption. If there are more wrestlers in one bracket than the other, then you'd have to explain in more detail how the draw is made.

Well sometimes you have a bracket with all 1/32 finale matches (only 16 wrestlers in that bracket) and the other bracket has mix of 1/32 and 1/64 matches. So some wrestlers go an extra match.

Each weightclass has 4 top seeds which garantees them they don't go that extra match but their position in which bracket they will be in is (suppose to be) random. So even if Kazakhstan has top seed together with a Russian or without them their placement in which bracket is suppose to be random.

 

[PeroK]

The real issue in many of these cases is to have some prior hypothesis of what might happen. The logic is this:

Although it's unlikely that Kazakhstan will avoid the Russians, it's not so unlikely that one particular country will be lucky and avoid the Russians. That country may then do better than expected, partly though their lucky draw. But, unless you have a separate reason to believe the draw was fixed in their favour, then this is just a regular part of a random draw.

You might say that Kazakhstan being the home team and getting a lucky draw is a coincidence; and it is. But that itself is not really enough.

Each weightclass has 4 top seeds which garantees them they don't go that extra match but their position in which bracket they will be in is (suppose to be) random. So even if Kazakhstan has top seed together with a Russian or without them their placement in which bracket is suppose to be random.

That changes things a lot. If a wrestler is a top-4 seed, then there is only a 1/3 chance of being drawn against the Russian: as a top 4 seed you have only 1/3 of the other seeds in your bracket. If we assume that the Kazakhstani and Russian wrestlers were all top-4 seeds, then there is about a 30% chance of avoiding the Russians in 8/10 brackets.

If some of the Kazakhstanis were seeded and some not, then it will be less than 30%. You'd have to do a specific calculation based on how many were seeded and how many not seeded.

The seeding makes all the difference. It makes it much more likely that two seeded wrestlers will avoid each other.

 

[Wi_N]

The real issue in many of these cases is to have some prior hypothesis of what might happen. The logic is this:

Although it's unlikely that Kazakhstan will avoid the Russians, it's not so unlikely that one particular country will be lucky and avoid the Russians. That country may then do better than expected, partly though their lucky draw. But, unless you have a separate reason to believe the draw was fixed in their favour, then this is just a regular part of a random draw.

You might say that Kazakhstan being the home team and getting a lucky draw is a coincidence; and it is. But that itself is not really enough.
That changes things a lot. If a wrestler is a top-4 seed, then there is only a 1/3 chance of being drawn against the Russian: as a top 4 seed you have only 1/3 of the other seeds in your bracket. If we assume that the Kazakhstani and Russian wrestlers were all top-4 seeds, then there is about a 30% chance of avoiding the Russians in 8/10 brackets.

If some of the Kazakhstanis were seeded and some not, then it will be less than 30%. You'd have to do a specific calculation based on how many were seeded and how many not seeded.

The seeding makes all the difference. It makes it much more likely that two seeded wrestlers will avoid each other.

it is fully possible that all 4 top seeds are in the same bracket. as i said their position in which bracket is random. this should not change anything.

 

[PeroK]

it is fully possible that all 4 top seeds are in the same bracket. as i said their position in which bracket is random. this should not change anything.

What's the point of being seeded? Apart from avoiding an extra 1/64 match? I've never heard of a seeding method where the seeds can all meet in the first round - in this case the 1/32 round.

Can you provide data to show a draw where all four seeds are in the same bracket?

There is simply no point in being seeded if the draw is subsequently random and being seeded makes no difference!

 

[Wi_N]

What's the point of being seeded? Apart from avoiding an extra 1/64 match? I've never heard of a seeding method where the seeds can all meet in the first round - in this case the 1/32 round.

Can you provide data to show a draw where all four seeds are in the same bracket?

There is simply no point in being seeded if the draw is subsequently random and being seeded makes no difference!

It's hard to prove cause in most weight classes one of the top seeds didn't show up and were replaced. But Kazakhstan only had 1 top seed so the placements of top seeds should not matter in this case. But let's say top seeds are evenly distributed across the brackets. still that leaves us 7/9 (according to you) or 7/7. But bare in mind when we are dealing with manipulation we can't be completely unbiased either. The match fixing happened in weight classes were it mattered. 57kg didn't cause Kazakhs had a top seed. 97kg and 125kg didn't matter either for them. The rest of the weight classes IT MATTERED and voila they faced no Russians.

edit: in 57kg Kazakhstan top seed was still not in the same bracket as the russian.

(just as a side note: wrestling is 50% strength and conditioning. That extra match matters A LOT.)