Unusual distributions became the "norm"

[FelicityR]

From Wikipedia:-

 

NORM

 

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector spaceexcept for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below.

A simple example is two dimensional Euclidean space R2 equipped with the "Euclidean norm" (see below). Elements in this vector space (e.g., (3, 7)) are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude.

A vector space on which a norm is defined is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space. It is often possible to supply a norm for a given vector space in more than one way.

 

Bridge is a game of mathematical possibilities, and, Unusual Distributions became this?

 

(Help! Can someone translate? :))

 

And YES! The forum members have commented on this again, and again, and again......Time to let sleeping dogs lie, or should that be, unusual distributions die?

[smerriman]

Yikes. mlbridge, you have a serious lack of understanding of statistics.

 

Your definition of "about 1%" seems to be 1.326% or lower.

 

On a single hand, the probability you are dealt a distribution which is "about 1%" or lower is about 7%; the sum of all of the low probabilities.

 

In a 12 board tournament, a given player will be dealt two or more "about 1%" hands 20% of the time. Not abnormal at all.

 

Over 24 boards, a given player will be dealt 5 or more "about 1%" hands about 2.3% of time.

 

The fact that it happened doesn't mean the deals aren't random.

 

But wait, this isn't what you measured. Why did you choose North? Was it perchance because you happened to notice that North had some distributional hands, and you didn't pick South because, whoops, that wouldn't have supported your theory?

 

In a 12 board tournament, the probability you will be able to find a player with 2 or more "about 1%" hands is about 58%. That's right it's actually more likely this will happen than it won't.

 

Over 24 boards, you'll be able to find a player with five or more 1% hands 9% of the time.

 

It would be extremely unusual if you *didn't* notice such things happening.

[zhoraster]

This might be the time for another public service announcement.

 

If you want to get better suit breaks and honors located where your finesses work, you need to subscribe to premium services.

 

A. Silver Premium membership - "normal" distributions, but finesses only work about 25%

B. Gold Premium membership - "normal" distributions, and finesses work 50%

C. Double Platinum membership - you always get the most even breaks, and finesses work 90%

There's a better one:

D. Ultimate (aka Solitaire) membership - always most even breaks, finesses always work: defenders' cards are rearranged if you misguess. In the case of overtricks, the contract is upgraded to the actual number of tricks taken (unless the contract is (re)doubled).

[hrothgar]

Nigel hits on the key issue here: The importance of formulating a hypothesis BEFORE choosing the data set.

 

From the sounds of things, mlbridge plays a fair number of robot tournaments.

 

Might I suggest the following:

 

1. Let's all (collectively) agree on a hypothesis that we want to test.

2. Having done so, lets agree how we plan to test this.

 

I recommend that we use next 20 tournaments that mlbridge players after we generate the hypothesis.

The sample is a bit small, but should be sufficient.

[aawk]

Another case of only remembering bad results and blaming the distribution for bad scores. If I learned one thing in bridge is that who gets angry at the table is the one who made the mistake.

[Joe_Old]

I love this thread. Americans are addicted to conspiracy theories, the more irrational, the better. Facts are irrelevant, to the deceived they are just part of the conspiracy. No one is ever going to convince someone like the poster that his worldview is skewed.

 

That's why conspiracy theories are central to the American psyche: they are a convenient excuse for failures that plainly can't be our own.

[miamijd]

Are you playing in free tourneys or paid ones?

 

Paid

[mlbridge]

Yikes. mlbridge, you have a serious lack of understanding of statistics.

 

Your definition of "about 1%" seems to be 1.326% or lower.

 

On a single hand, the probability you are dealt a distribution which is "about 1%" or lower is about 7%; the sum of all of the low probabilities.

 

In a 12 board tournament, a given player will be dealt two or more "about 1%" hands 20% of the time. Not abnormal at all.

 

Over 24 boards, a given player will be dealt 5 or more "about 1%" hands about 2.3% of time.

 

The fact that it happened doesn't mean the deals aren't random.

 

But wait, this isn't what you measured. Why did you choose North? Was it perchance because you happened to notice that North had some distributional hands, and you didn't pick South because, whoops, that wouldn't have supported your theory?

 

In a 12 board tournament, the probability you will be able to find a player with 2 or more "about 1%" hands is about 58%. That's right it's actually more likely this will happen than it won't.

 

Over 24 boards, you'll be able to find a player with five or more 1% hands 9% of the time.

 

It would be extremely unusual if you *didn't* notice such things happening.

 

I choose north because that's my pard. Did not go thru in advance all four to see which one have the most lowest percentage distribution.

 

I may not be a stat whiz. But if someone tells me that the probability of one quarter of the hands (with odds close to 1%) dealt to one player is somewhat normal, I would certainly question that.

[strags]

If you play the Robot Tournaments almost exclusively, as I do, and you suspect that significantly more than half of your finesses lose in those games...you're correct. In fact, one longtime friend and I have a running joke where we will often mention whimsically in our conversations, "Hey, did I tell you that a finesse worked for me on BBO the other day?"

 

But it has nothing to do with faulty software or unrealistic deal probabilities. It's simply a function of two perfectly innocent factors:

 

• The "Best Hand for South" format
  
• The opening-lead tendencies of the BBO bots
  

 

Put simply: if you're playing a Robot Tournament, you're sitting South and are dealt the best hand at the table. Therefore, you are more likely than your robot North partner to (a) become declarer, and (b) hold high-card tenace positions like AQ or KJ in which you wish to take a finesse.

 

Ergo, you will be taking these finesses disproportionately into the West hand. And, West will disproportionally happen to be the opening leader.

 

BBO bots are notoriously passive leaders. That much, I hope everyone agrees on. This is hardly surprising, since there have been a raft of books published in the last 10 years that have analyzed opening lead probabilities and concluded that passive is usually better. Indeed, the one counterexample those books have shown is to lead very aggressively (i.e., away from a king or queen) in a side suit against a small suit slam. And, in my experience, that's also the one time a BBO bot can be counted on to have led away from a royal.

 

So...in BBO Robot Tournament games, West is usually on opening lead. West will lead very passively, usually from a suit in which he holds no honor above a 10. You will more often finesse (in some other suit) through East and into West because of the "Best Hand" format. Ergo, it's more likely -- I'd wager between 55% and 60% -- that these finesses will lose. There's nothing sinister about this. It's just a consequence of the unusual form of bridge we enjoy on BBO.

[smerriman]

I may not be a stat whiz. But if someone tells me that the probability of one quarter of the hands (with odds close to 1%) dealt to one player is somewhat normal, I would certainly question that.

As mentioned, the probability at least one of the four players has 5 of 24 "1%" hands (by your definition) is 9%. It's not "normal", but it will happen regularly enough, so having it happen definitely doesn't mean anything odd is going on.

 

If you don't understand why, try this calculator:

 

https://stattrek.com/online-calculator/binomial.aspx

 

The probability of a '1%' hand is 7%, since there are so many '1%' distributions. Fill in 0.07, 24 trials, 5 successes, and you'll see P(X>=x) = 2.3%.

 

Then for this to apply to all four players is 1 - (1-0.023)^4 = 9%.

 

(Not this isn't 100% accurate due to the lack of independence, and my calculations earlier were more accurate, but they're close enough).

[barmar]

Those of you having trouble believing the probabilities that smerriman is describing might want to google "birthday paradox".

[mlbridge]

Okay - so I did a different analysis looking at all four hands. From Googling, on a particular hand, it seems that the probability of any player having been dealt a void is about 5.1% So for 12 boards, one would expect less than one instance occurring for the group. Just finished Robot rebate tourney #4584 - 12/14/2019.

 

W - Two voids, boards 7 & 9

 

S - One void, board 12

 

N - One void, board 9.

 

So on board 9, two players were dealt voids.

 

Also, W had the most volatile distributions. Of the other 10 boards without voids, W was dealt a singleton 6 times. So on 12 boards, W had 2 voids and 6 singletons.

[hrothgar]

If you play the Robot Tournaments almost exclusively, as I do, and you suspect that significantly more than half of your finesses lose in those games...you're correct. In fact, one longtime friend and I have a running joke where we will often mention whimsically in our conversations, "Hey, did I tell you that a finesse worked for me on BBO the other day?"

 

But it has nothing to do with faulty software or unrealistic deal probabilities. It's simply a function of two perfectly innocent factors:

 

• The "Best Hand for South" format
  
• The opening-lead tendencies of the BBO bots
  

 

 

Interesting piece

 

thanks for posting it

[johnu]

The quality of the trolls is really declining...

 

Can we send this one back and get something better?

I find it interesting that pink flag signed up for the forums on the day of their first 2 and only posts which happen to be in this thread.

[mlbridge]

So after #4584, I played in order:

 

#4769 - 1 hand with a void

 

#5170 - 4 different hands with a void.

 

So in 36 boards, 8 different hands had a void and one hand had 2 voids.

 

Explain to me how that is random.

[hrothgar]

So after #4584, I played in order:

 

#4769 - 1 hand with a void

 

#5170 - 4 different hands with a void.

 

So in 36 boards, 8 different hands had a void and one hand had 2 voids.

 

Explain to me how that is random.

 

The basic problem here is that you fundamentally misunderstand basic concepts like sample sizes and the requirement that one formulates hypothesizes BEFORE looking at the data set rather than afterwards.

 

If you go and look at a small data sample, you'll almost inevitably be able to find some way in which the observed data does not match the expected.

 

Case in your original posting, you were complaining that there were too many singletons.

 

I believe the quote was

 

More startling is if one looks closer at the single suited hands. One have less than 1% of occurring and 2 have 1.3% chance. So of the 12 hands, N was dealt 3 hands were about a 1% chance of occurring.

 

Now, after looking at a new set of boards you are advancing the claim that the the hand generator is producing too many voids.

 

If you want folks to take you seriously, make a specific testable claim about the hands that BBO generates and we can test these against some future day's worth of tournament data.

[smerriman]

So after #4584, I played in order:

 

#4769 - 1 hand with a void

 

#5170 - 4 different hands with a void.

 

So in 36 boards, 8 different hands had a void and one hand had 2 voids.

 

Explain to me how that is random.

You're making similar mistakes as last time.

 

The probability a *given player* (eg South) has a void is about 5%.

 

The probability at least one of the four players for a given deal has a void is about 20%.

 

Over 36 boards, the probabilities of seeing a total of x voids (approximately):

 

0: 0.1%

1: 0.5%

2: 1.8%

3: 4.2%

4: 7.8%

5: 11.4%

6: 13.9%

7: 14.8%

8: 13.6%

9: 11.0%

10: 8.2%

11: 5.5%

12: 3.4%

13: 1.9%

14: 1.0%

15: 0.4%

 

All of 5, 6, 7, 8 and 9 voids occur over 11% of the time, so aren't unusual in the slightest.

[mlbridge]

So continuing on, next tourney I played in is #5170.

 

There were 4 boards with a least one void. One of the boards have both opp with a void.

 

By the way, I do not noticed such distributions when playing in our imps tournaments. Only these robot rebate tourneys.

[smerriman]

Now you're just making things up. I checked #5170, and there wasn't a single board that had two voids.

 

You had a typo; you meant #5519.

 

4 voids is not uncommon; it happens on average about once every four tournaments, and you're finding it surprising.

 

Even 5 voids isn't strange; the chance you've seen a 5 void tournament over 4 tournaments is about 36%.

 

And you only need to play 3 tournaments to make it more likely that you've seen a deal with 2 voids than you haven't.

 

You're just making yourself look silly now.

[smerriman]

To top things off, guess how many voids you saw in the two Robot Rebate tournaments before the two you started reporting?

 

1 and 1.

 

Below average.

 

Bet you didn't pay much attention to those.

[mlbridge]

All I know is in 48 boards, 4 tourneys, I have 12 boards with voids. Two of them have 2 voids.

 

According to your math, I should have about 20% of 48 boards = 9.6. I think 12 is much greater than 9.6, especially on a percentage basis. And that's not including the extra 2 voids.

 

You also indicate that 4 voids happen about once in every 4 tournaments. In my case, it happened 3 out of 4.

 

My analysis also includes the tourney with just 1 void.

 

I am not sure how many here play in the robot rebate tourneys. My personal experience is that those hands are far more distributional than the regular tourneys.

[smerriman]

According to your math, I should have about 20% of 48 boards = 9.6. I think 12 is much greater than 9.6, especially on a percentage basis.

This is completely incorrect statistically. With such a tiny sample, a standard 95% confidence interval would be [4.16,15.03]. You would only be able to claim - and of course, not prove, since it can still happen 5% of the time - there is evidence of non-randomess if your average fell outside this range. Anything inside the range fits the null hypothesis that there is no bias.

 

You also indicate that 4 voids happen about once in every 4 tournaments. In my case, it happened 3 out of 4.

Again, completely inconsequential statistically.

 

My analysis also includes the tourney with just 1 void.

I was referring to the two prior to all of the ones you've listed. As of course, you only started measuring voids because you suddenly noticed a lot of them in one tournament; before that you were focusing on something unrelated (and also incorrect). Choosing when to start measuring heavily biases the results.

[hrothgar]

Not sure why I'm bothering, but...

 

For kicks and giggles, I ran the following command in R

 

> 1 - pbinom(3, size=12, prob=.051*4)

[1] 0.2161656

 

In English, this tells me the probability that I will see 4+ voids in a tournament of 12 boards.

I am calculating this by looking at the probability that I will generate 0, 1, 2, or 3 voids and then subtracting this from 1

 

Note, I am fudging things by assuming that the probability that you have a void on a board is 4 X (the probability that you have a void in one hand which isn't quite true

Like smerriman says, this is slightly more than 20%

 

Next, I ran the following command

 

> rbinom(1000, 1, .2161656)

 

This command simulates 1000 virtual tournaments and codes a 0 if you had between 0-3 voids and a 1 if you observed 4+ voids.

 

Note how frequently you see cases where three out of four tournaments have 4+ voids.

There was even one case in which 5 tournaments in a row had 4+ voids.

Weird ***** happens all the time

 

[1] 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

[38] 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0

[75] 0 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0

[112] 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0

[149] 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0

[186] 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

[223] 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0

[260] 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1

[297] 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1

[334] 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0

[371] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0

[408] 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

[445] 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 1

[482] 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

[519] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

[556] 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0

[593] 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0

[630] 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0

[667] 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0

[704] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1

[741] 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0

[778] 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

[815] 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0

[852] 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1

[889] 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1

[926] 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0

[963] 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 0

[1000] 0

[barmar]

I just checked the last 90 days of mlbridge's hands in the database used by myhands (so it doesn't include daylongs and challenges).

 

There were 4055 hands. South had 176 voids, West had 199 voids, and North had 205 voids. We don't store East's hand in the database, software that displays hands calculates it from the other 3 hands, so I couldn't easily get the number of voids there. 37 of the hands had voids in multiple seats.

 

The best-hand feature is almost certainly the reason why South has fewer voids than the other seats, since a void contributes 0 HCP (BH doesn't add anything for distribution).

 

Someone with better statistics knowledge can tell us how far off this is from expectations.

[smerriman]

Taking best hand into account, my calculations for 95% confidence are:

 

South: (174.1,228.3)

Other seats: (182.1,237.4)

Multiple voids amongst South/West/North only: (28.7,53.8)

 

All those numbers lie in these ranges (if anything slightly below average, but not significantly enough).