An Observed Extreme Probability Event

[Hornbein]

At the 2005 World Bridge Championships in Estoril a hand was dealt with QT8642 in two different suits and with the remaining card even. The odds of this happening by pure chance are one in 531 trillion. We can say that AQT864 and KJ9753 would form an equivalent class of "hands spaced by two", reducing the odds to one in 177 trillion.

I'm checking with experts, but it seems to me that this deal must have been done with real cards that had been hand shuffled from a brand new deck, in which all the cards are strictly ordered. It's impractical to truly randomize cards that way. I believe that tournament bridge has since gone to mathematical randomization, which means this won't happen again. When this change was made bridge players noticed a difference (and complained about it, as it threw them off).

As further evidence there once was a tournament deal from a brand new deck that hadn't been shuffled at all. The players noticed and had the deal invalidated.

 

[Dale]

The odds of this happening by pure chance are one in 531 trillion.

You need to be pretty clear what is meant by "this happening". There are only 635 billion bridge hands (52 choose 13), so I don't know what "this happening" they are talking about.

The odds of being dealt any given bridge hand is 1 in 635 billion. So in that sense every single game is four "extreme probability events" all happening together.

The odds of being dealt an all even bridge hand is closer to 1 in 8000.

 

[FactChecker]

I know nothing about bridge, but there is something you should be careful of when you think that a random result is unreasonably improbable. You should always determine or estimate the total probability of getting a result with that probability or lower.
For instance, suppose you draw a random real number between 0 and 1. The probability of getting exactly that number is extremely small. But you must get some number and they all have equally small probabilities. So the small probability of that particular number proves nothing.
In your example, you should figure out the total probability of all the deals that would have chances of one in 531 trillion or less. It is not enough to just include that other hand KJ9753.

 

[Hornbein]

The odds against this particular hand are a product of:

The chance that you got dealt six cards in two suits and one in the other. That's about one in one thousand.
This is multiplied by the chance that only six ranks are present in such a 6610 hand. That's one in 156156. (This implies that the two suits have identical ranks).
This is multiplied by the chance that such suits are spaced with a difference of two ranks from each card to the next. This is one in 286.
This is one in 44.7 billion. So I did double count and get it wrong, but "fortunately" the result is still such long odds that the conclusion remains supported, that in 2005 the deals weren't completely random.

Sure, you can say all deals are equally likely so who cares but I assure you bridge players notice some hands over others and there is a great deal of agreement as to which hands are the most noticeable.

 

[Hornbein]

One may observe that there is a whole class of hands of various forms that are equally or even more amazing. The specification of such a class is a matter of opinion though. I'm not going to attempt that but would only guess that this reduces the odds by a factor of a hundred or a thousand or so, which puts the odds of this ever having occurred within the range of possibility.

If you however quite reasonably restrict to tournament hands for which we have trustworthy records and assurance the deals weren't rigged then I suppose the odds against it having ever happened become once again large.

 

[Dale]

I assure you bridge players notice some hands over others and there is a great deal of agreement as to which hands are the most noticeable.

Sure. But the odds of some “noticeable” hand is much higher than the odds of specifically a 6610 all even hand.

That the actual noticeable hand happened to be some specific noticeable hand is not particularly impressive unless it was specifically predicted in advance. Otherwise it is really just any noticeable hand and the odds given should be the odds of a general noticeable hand rather than the odds of a specific noticeable hand.

The issue is that this type of retrospective astonishment commonly vastly overstated the surprisal by not clearly specifying the actual class of surprising events. This is related scientifically to p-value inflation and the replicability crisis.

 

[Hornbein]

Sure. But the odds of some “noticeable” hand is much higher than the odds of specifically a 6610 all even hand.

That the actual noticeable hand happened to be some specific noticeable hand is not particularly impressive unless it was specifically predicted in advance. Otherwise it is really just any noticeable hand and the odds given should be the odds of a general noticeable hand rather than the odds of a specific noticeable hand.

The issue is that this type of retrospective astonishment commonly vastly overstated the surprisal by not clearly specifying the actual class of surprising events. This is related scientifically to p-value inflation and the replicability crisis.

I already addressed this.

 

[Hornbein]

Doing the calculation yet more carefully -- I have always been sloppy with this --
I figure a 6610 with the ranks separated by two and a matching singleton occurs in one out of 2.94 billion hands.

There 3 ways a six card suit can have all ranks separated by two ranks. There are 6 combinations of two of the four suits. For each such hand there are 2*6=12 qualifying singletons. 3*6*12=216 hands in this class. There are 635 billion possible bridge hands. Divide by 216 and you get 2.94 billion.

 

[Dale]

I already addressed this.

Partially. As you mentioned, there are a lot of other surprising hands. Some of those are exceptionally rare, like 13 all of one suit. There are only four of those, so including them doesn’t change the odds much. Let’s be more credulous than the factor of 100 to 1000 you suggested and say it is only a factor of 3. That brings us down from 1 in 3 billion to 1 in a billion. This isn’t actually a statement about bridge hands, but about surprise. If we see any 1 in a billion hand we will be very surprised. A 1 in 100 million hand is cool, but won’t be considered very surprising.

Nobody (as far as I can tell) predicted that this specific player would get the surprising hand, so it wouldn’t be 1 in a billion, but 4 in a billion.

There was also no prediction that it would be a specific deal, so you would have been equally surprised to see the result in any deal in the game. There are roughly 25 deals per game, so that is now 100 in a billion.

You would have also been equally surprised if it had happened in a different game in that tournament. I tried to find out how many games are in a tournament, but couldn’t after a few minutes so I guess maybe 20. That is now 2000 in a billion.

There is something special about a world level competition, so we won’t include lower level games, but there is nothing special about 2005. We would have been equally surprised to see this in world level competition in any of say the last 50 years. So that brings us to 100000 in a billion, or 1 in 10 thousand.

That is certainly “significant” in the usual 1 in 20 sense used by the scientific community. But it is not nearly as extremely surprising as we thought. This is known as the “multiple comparisons” issue and is what turns a 1 in a billion chance to merely a 1 in 10 thousand chance. This happens all the time in real research.

the result is still such long odds that the conclusion remains supported, that in 2005 the deals weren't completely random

The observation of one low probability event never provides evidence supporting a conclusion of non-randomness. Nor even multiple low probability events.

Again, from a probabilistic point of view each hand is a 1 in 635 billion event. And four such unlikely events occur in each round.

To test for randomness you actually want to divide your entire sample space into two mutually exclusive groups that have 1 to 1 odds. Then, you look at your “runs” and if you have too many or two few then it is non random.

So, for example, if you wanted to see if the 2005 tournament were random you might put all rounds where the ace of spades was in north or south as group A and all rounds where it was in east or west as group B. Then you could sort the rounds by time when the dealing started. You would count the number of “runs”. So AABABBB would be 4 runs. Then you could apply the Wald Wolfowitz runs test to provide evidence against the assumption of randomness.

 

[DaveC426913]

Sure. But the odds of some “noticeable” hand is much higher than the odds of specifically a 6610 all even hand.

That the actual noticeable hand happened to be some specific noticeable hand is not particularly impressive unless it was specifically predicted in advance. Otherwise it is really just any noticeable hand and the odds given should be the odds of a general noticeable hand rather than the odds of a specific noticeable hand.

This is very similar to a much simpler phenomenon (and thus, perhaps, easer to analyze) occurring in lottery numbers.

A Canadian lottery has seven numbers being picked from 1 to 49.

My brother was convinced that betting on 1,2,3,4,5,6,7 was folly, since it was astronomically unlikely to turn up. I pointed out to him that any seven picked numbers, (such as, say, 5,8,14, 22,23,31,48) are exactly as unlikely.

 

[BWV]

This is very similar to a much simpler phenomenon (and thus, perhaps, easer to analyze) occurring in lottery numbers.

A Canadian lottery has seven numbers being picked from 1 to 49.

My brother was convinced that betting on 1,2,3,4,5,6,7 was folly, since it was astronomically unlikely to turn up. I pointed out to him that any seven picked numbers, (such as, say, 5,8,14, 22,23,31,48) are exactly as unlikely.

As winning lottery tickets split the pot, there would be small advantage to picking consecutive numbers if a sufficiently large number of players select numbers manually and avoid consecutive numbers. Its also possible that the random quick pick may be tweaked to avoid consecutive numbers given the players bias

 

[BWV]

I thought competitive bridge was all duplicate (everyone plays the same pre-selected hands)

 

[mathman]

I know nothing about bridge, but there is something you should be careful of when you think that a random result is unreasonably improbable. You should always determine or estimate the total probability of getting a result with that probability or lower.
For instance, suppose you draw a random real number between 0 and 1. The probability of getting exactly that number is extremely small. But you must get some number and they all have equally small probabilities. So the small probability of that particular number proves nothing.
In your example, you should figure out the total probability of all the deals that would have chances of one in 531 trillion or less. It is not enough to just include that other hand KJ9753.

the probability of drawing a particular real number = 0. You can't use uncountable sets (all reals between 0 and 1) to get probability.

 

[PeroK]

Doing the calculation yet more carefully -- I have always been sloppy with this --
I figure a 6610 with the ranks separated by two and a matching singleton occurs in one out of 2.94 billion hands.

There 3 ways a six card suit can have all ranks separated by two ranks. There are 6 combinations of two of the four suits. For each such hand there are 2*6=12 qualifying singletons. 3*6*12=216 hands in this class. There are 635 billion possible bridge hands. Divide by 216 and you get 2.94 billion.

Let's assume that the probability of drawing a special hand is about one in a billion. I could probably come up with 1000 to 10000 special hands. The probability of getting something special is probably nearer one in a million. That's one every 250,000 bridge games. If there are 10,000 bridge games every year That's one special hand every 25 years.

You had to go back nearly 20 years to find one.

 

[PeroK]

the probability of drawing a particular real number = 0. You can't use uncountable sets (all reals between 0 and 1) to get probability.

It's physically impossible to draw with equal likelihood from more than a finite number of options.

 

[PeroK]

Let's assume that the probability of drawing a special hand is about one in a billion. I could probably come up with 1000 to 10000 special hands. The probability of getting something special is probably nearer one in a million. That's one every 250,000 bridge games. If there are 10,000 bridge games every year That's one special hand every 25 years.

You had to go back nearly 20 years to find one.

To tidy up these numbers. To get to one in million we would need 635,000 special hands. That would be, say, 1000 special hands, each with several hundred variations in terms of suits and suboptions. In the example quoted the spare card could have been anything from one of the other suits. That's 26 possibilities. Then, there were 6 combinations of two suits. That's 150 possibilities. That particular hand has a probability of about one in four billion.

PS then we need about 1,000 special hands.

 

[Hornbein]

So, did that bridge hand with only even cards an artifact of the shuffling procedure or was it just luck?

Prestidigitators have long studied shuffle tricks, the famous reference being Chris Marlo's 1953 "Faro Notes." A perfect shuffle is dividing the deck into two equal halves then alternating cards from each half. Eight consecutive perfect shuffles will return a deck of cards to its original order. Decks of cards are always sold with the cards in the same order. Five perfect shuffles will partition a brand new deck into half even cards and half odd cards. My thesis is that in the 2005 Bridge World championship a brand new deck was shuffled seven times, an unusually zealous attempt at randomization. If the shuffles are perfect this will generate a deck where there is a sequence where every fourth card is an even card. When four hands are dealt this is will result in a hand with all even cards, such as was seen at the table.

The shuffles didn't have to be perfect, just close enough. There are also numerous sequences where errors cancel one another. Believe whatever you want, but I'll take this over the one in 2.4 billion shot that assumes a uniform distribution.

 

[Dale]

So, did that bridge hand with only even cards an artifact of the shuffling procedure or was it just luck?

It could have been either. To establish non-randomness requires different data.

I'll take this over the one in 2.4 billion shot that assumes a uniform distribution.

It is closer to a 1 in 10000 shot, as described above.

 

[FactChecker]

the probability of drawing a particular real number = 0.

Good point. I didn't want to get into the details. I consider 0 to be "extremely small", as I said.

You can't use uncountable sets (all reals between 0 and 1) to get probability.

I'm not sure that I understand you here. A lot of probability theory deals with uncountable sets, as long as they are measurable. If you randomly draw a number, X, from the interval [0,1], you can say that the probability it is in [0,1] is 1. If it is a uniform distribution, you can say that the probability it is in [0., 0.5] is 1/5.

UPDATE: I must be slow. I guess you are talking about nonzero probabilities on the individual points of an uncountable set. I agree with you there.

 

[PeroK]

So, did that bridge hand with only even cards an artifact of the shuffling procedure or was it just luck?

Prestidigitators have long studied shuffle tricks, the famous reference being Chris Marlo's 1953 "Faro Notes." A perfect shuffle is dividing the deck into two equal halves then alternating cards from each half. Eight consecutive perfect shuffles will return a deck of cards to its original order. Decks of cards are always sold with the cards in the same order. Five perfect shuffles will partition a brand new deck into half even cards and half odd cards. My thesis is that in the 2005 Bridge World championship a brand new deck was shuffled seven times, an unusually zealous attempt at randomization. If the shuffles are perfect this will generate a deck where there is a sequence where every fourth card is an even card. When four hands are dealt this is will result in a hand with all even cards, such as was seen at the table.

The shuffles didn't have to be perfect, just close enough.

It changes things if you have a specific, potential scenario that produces that particular hand. Then, that specific hand is potentially predictable.

If, however, the cards were unshuffled, then wouldn't the other three hands show a similar pattern?

 

[PeroK]

For example, if the first hand was A, 5, 9, K of spades; and the second hand was 2 6, 10 of Spades and A of Hearts etc., then the probability is almost certain that the deck was unmixed. That must have happened many times, where players started a card game with a fresh deck and forget to mix it.

If there are other possibilities for partial mixing or mixing gone wrong, then those need to be assessed using the estimated probability that such partial mixing took place.

 

[FactChecker]

For example, if the first hand was A, 5, 9, K of spades; and the second hand was 2 6, 10 of Spades and A of Hearts etc., then the probability is almost certain that the deck was unmixed. That must have happened many times, where players started a card game with a fresh deck and forget to mix it.

If there are other possibilities for partial mixing or mixing gone wrong, then those need to be assessed using the estimated probability that such partial mixing took place.

Furthermore, given these hands were dealt, Bayes' Theorem can be used to imply whether or not the deck had been randomly shuffled. That might be what the OP was driving at. In the analysis, we would still need to be careful to include other equally unlikely hands which could falsely imply no shuffling even if the deck had been shuffled. IMO, it's a tricky analysis.

 

[BWV]

As I mused earlier, the whole thread is based on a false premise. Bridge championships, including the 2005 World Championship do not involve random deals - to measure the skill of the contestants, all teams play the same preselected set of hands, so perhaps a random deal was selected for the board, or perhaps the odd arrangement was desired by the organizers

https://www.bridgehands.com/Tournaments/WBF/2005_World_Team_Championship/bul_05.pdf

 

[BWV]

But there is this from the 2008 World Series of Poker;

In a stunning statistical improbability, Justin Phillips knocked out Motoyuki Mabuchi in the Main Event. Phillips held a Royal Flush, while Mabuchi held quad aces. One of the broadcasters, Lon McEachern, mentioned on air that the chances of such a showdown occurring were 1:2.7 billion.

https://en.m.wikipedia.org/wiki/2008_World_Series_of_Poker#:~:text=In a stunning statistical improbability,occurring were 1:2.7 billion.

 

[hmmm27]

Phillips held a Royal Flush, while Mabuchi held quad aces.

Umm.... so, where did the spare ace come from ?

 

[BWV]

Umm.... so, where did the spare ace come from ?

It’s Texas Holdem, no? One or more aces was in the flop

 

[BWV]

here is the hand

 

[Nugatory]

I thought competitive bridge was all duplicate (everyone plays the same pre-selected hands)

They are not preselected but the same deal is played more than once. In pairs events the cards are randomly dealt by computer (and an algorithm that does this correctly, in the sense that all possible arrangements are equally probable, is surprisingly difficult) while in team games the cards are hand-shuffled except at the highest levels of competition.

 

[Nugatory]

I'm checking with experts, but it seems to me that this deal must have been done with real cards that had been hand shuffled from a brand new deck, in which all the cards are strictly ordered.

It’s the other way around, imperfect hand shuffling of a new deck produces anomalously flat hands that are closer to the expectation values (3.25 cards in each suit, one card of each rank) then we find when the cards are dealt randomly. Because of the way the cards are placed on the table during play the same thing happens even when we aren’t starting with a new deck.

 

[FactChecker]

It’s Texas Holdem, no? One or more aces was in the flop

So there can be up to 4 Royal flushes simultaneously in the hands. At the risk of beating a dead horse, I repeat that the rarity of any single event should be considered in light of all the events that are equally or less likely.

 

[gmax137]

So, did that bridge hand with only even cards an artifact of the shuffling procedure or was it just luck?

Prestidigitators have long studied shuffle tricks, the famous reference being Chris Marlo's 1953 "Faro Notes." A perfect shuffle is dividing the deck into two equal halves then alternating cards from each half. Eight consecutive perfect shuffles will return a deck of cards to its original order. Decks of cards are always sold with the cards in the same order. Five perfect shuffles will partition a brand new deck into half even cards and half odd cards. My thesis is that in the 2005 Bridge World championship a brand new deck was shuffled seven times, an unusually zealous attempt at randomization. If the shuffles are perfect this will generate a deck where there is a sequence where every fourth card is an even card. When four hands are dealt this is will result in a hand with all even cards, such as was seen at the table.

The shuffles didn't have to be perfect, just close enough. There are also numerous sequences where errors cancel one another. Believe whatever you want, but I'll take this over the one in 2.4 billion shot that assumes a uniform distribution.

I don't know about Bridge, but in casinos I often see the dealers "washing" the cards, ie, mixing them up the way a kid who can't shuffle would.

 

[Stephen Tashi]

Like many probability questions on the forum that touch on real life situations, this discussion doesn't make much reference to mathematical statistics !

It's interesting to look at the question from the point of view of hypothesis testing - not that hypothesis testing clarifies the question, but rather that the question reveals a lot about the subjective nature of hypothesis tests.

It often happens in hypothesis testing that any particular observed outcome (e.g, any deal) has a very small probability - smaller than the traditional alpha values used in hypothesis tests. So the "acceptance region" for accepting the null hypothesis ( that the deal was randomly dealt) is defined in terms of some larger set of deals. (Some statisticians violently object to the terminology "accepting the null hypothesis". They prefer to look at the complement of the acceptance region, which is the rejection region.)

To specify the acceptance region (which the OP calls an equivalence class of hands) we must either enumerate all the deals in the set or state some some characteristic of a deal that determines whether it is in the set. The probability that a hand falls into an acceptance region can vary widely depending on how that acceptance region is defined - as illustrated by previous posts in this thread. Although statisticians want to be unbiased, it is hard to resist the temptation of trying out various acceptance regions containing the observed result to see whether the associated probabilities please one's intuition.

The textbook treatment of acceptance regions is restricted to "one-tailed" or "two-tailed tests". This is the case when the outcome is a numerical measurement. Intuition that those acceptance regions are good is convincing, but proof that they are optimal (in some sense) is hard to appreciate. It has to do with the "power" of a statistical test.

When it comes to defining the power of a statistical test on deals of cards, I don't see any objective way of doing it. In the case of a numerical measurement, one graphs a function of the difference between the two scalars. For deals of cards, does any (possibly multivariable) function suggest itself ?