An unusual occurrence?

[WellSpyder]

(A bridge-related query, but really about statistics, hence the post in the Water Cooler.)

 

4441 hands (which I believe may be described as 4x1 on the other side of the Atlantic) are well known for being tricky to bid, but fortunately they aren't very common, with the result that in one or two systems I play they are more or less ignored and you are just expected to do the best you can. However, I played a 24-board match last night, and towards the end of the first half I realised that I was holding at least my 3rd 4441 hand of the evening. And there were two more in the second half. Hand records weren't available, and I can't remember for sure whether there had been any more in the first half before I started noticing their repeated appearance.

 

Anyway, assuming I "only" held five 4441 hands over the course of the 24 boards, I found myself wondering how unusual an occurence this was. According to my calculations, the chances of a given hand having a 4441 shape are just under 3%, so the expected number over the course of 24 boards is 0.72, but I wonder whether anyone can quickly work out the probability of at least 5 such hands. (I recall learning back in the dim and distant past - perhaps 35 years ago - about using a Normal approximation to a Binomial distribution, which might be one way of tackling this, but I've no real idea of how valid the approximation might be for these sort of numbers....)

[MickyB]

According to http://web.inter.nl.net/hcc/M.A.F/c30-37e.htm:

 

P (being dealt a 4441) = 2.993%

 

There are binomial calculators online, I used http://stattrek.com/online-calculator/binomial.aspx

 

P (precisely five 4441s in 24 attempts) = 0.0573%

P (at least five 4441s in 24 attempts) = 0.0634%

[WellSpyder]

Thanks, MickyB! Looks like exactly the answer I was looking for.

 

So assuming I play a 24-board session once a week then I should expect to get 5 or more 4441 hands in a session approximately every 30 years.....

[helene_t]

yesterday I had some pretty dull hands:

 

4x 5431

1x 5422

2x 4333

1x 6520

1x 4441

2x 5322

13x 4432

 

With 4 of the five 54xx hands being 11-15 HCPs with 4♠5♥ it was a good eveing for Flannery (except that on two of those four deals someone else opened in front of me).

 

But I think the probability of observing at least one perculiar feature of the statistics at any one evening is close to 1 :)

[Trinidad]

But I think the probability of observing at least one perculiar feature of the statistics at any one evening is close to 1 :)

 

IMO, one part of understanding statistics is to realize that it is unexpected when everything is as expected.

 

Apart from being helpful in predicting what is expected to happen, statistics are also pretty good at predecting just how much unexpected will happen. Unfortunately, statistics can't tell what unexpected is expected to happen.

 

Rik

[Vampyr]

yesterday I had some pretty dull hands:

 

4x 5431

1x 5422

2x 4333

1x 6520

1x 4441

2x 5322

13x 4432

 

With 4 of the five 54xx hands being 11-15 HCPs with 4♠5♥ it was a good eveing for Flannery (except that on two of those four deals someone else opened in front of me).

 

But I think the probability of observing at least one perculiar feature of the statistics at any one evening is close to 1 :)

 

 

Quite. Perhaps Mike or someone else would be so kind as to calculate the probability of your holding that combination of shapes.

[billw55]

I must be very lucky, every hand I pull out of the board is a one in 600 billion chance.

[PeterAlan]

Quite. Perhaps Mike or someone else would be so kind as to calculate the probability of your holding that combination of shapes.

As a first approximation, much less than 1 in 1000, since (a) those 7 shapes cover 74.7588% of all hands, so (b) the probability of 24 hands containing only those shapes (irrespective of individual shape frequency) = 0.747588^24 < 0.1%. To go into further detail would of course be very obsessive and boring ... ;)

[ahydra]

I had a 61-hand streak (at one club, over 3 weeks) of 14 or fewer HCPs. For one hand it's 85.82% apparently, so the odds of such a streak is 1 in 11,248.

 

Still, at least there were some fun hands in there. Thirteen 4432s in a night sounds like something that would cause one to fall asleep at the table.

 

ahydra

[helene_t]

I was told this anecdote is true but even if not it is funny anyway. Two mathematicians sit in a taxi. One of them says: "Look what an interesting license plate in front of us! It is the sum of three consecutive cubes! Very unusual!"

 

The other mathematician says: "So what? There is an infinity of unusual numbers. Nothing special about them".

 

"How do you know?"

 

"Suppose that there is only a finite number of unusual natural numbers. Now calculate the sum of all those. That will clearly not be in the set. However, being the sum of all unusual natural numbers is clearly an unusual feature!. Q.E.D."

[mycroft]

Ah yes, the "first uninteresting integer" problem extended to combinations. I've always liked that one.

[Zelandakh]

The other mathematician would have replied that there is not an infinite number of unusual numbers with the right number of digits to fit on a number plate. There is an infite number of prime numbers but these are still rather interesting. There are probably an infinite number of Fibonacci primes but they are certainly unusual and we only know of 33 for certain. Similarly, there is a finite amount of paint in the world but that does not make watching paint dry special or interesting. From a mathematical point of view, I would think that infite sets are generally more interesting/special than finite ones. But perhaps the guy was just trying to be a smartarse...

[barmar]

The second mathematician is right. If you're given a number, you can probably always find some attribute that makes it a member of a very infrequent class. In fact, if you couldn't come up with anything particularly noteworthy, that in itself would make it interesting. It's a set theoretical oxymoron: it's interesting because there's nothing interesting about it.

 

As with statistics, things are only really remarkable if you predict them ahead of time, rather than ex post facto. It's really hard to predict who will win the lottery, but it's a good bet that someone will -- on the day after, there's not much point in saying that the person who won it had an infinitessimal chance.

[PeterAlan]

It is the sum of three consecutive cubes! Very unusual!"

The set of natural numbers that are the sum of 3 consecutive cubes (of positive integers) is clearly countably infinite, since it can be put in 1-1 correspondence with the set of positive natural numbers in an obvious way: the first is [1^3 + 2^3 + 3^3], the second is [2^3 + 3^3 + 4^3] and, generally, the nth is [n^3 + (n+1)^3 + (n+2)^3].

 

This method generalises to show that any set of natural numbers defined in a suitably similar well-defined "unusual property" way is countably infinite (provided they are demonstrably distinct from one another).

[FM75]

I must be very lucky, every hand I pull out of the board is a one in 600 billion chance.

LOL - you must always pull boards that have exactly 13 cards.

[FM75]

(A bridge-related query, but really about statistics, hence the post in the Water Cooler.)

 

4441 hands (which I believe may be described as 4x1 on the other side of the Atlantic) are well known for being tricky to bid, but fortunately they aren't very common, with the result that in one or two systems I play they are more or less ignored and you are just expected to do the best you can. However, I played a 24-board match last night, and towards the end of the first half I realised that I was holding at least my 3rd 4441 hand of the evening. And there were two more in the second half. Hand records weren't available, and I can't remember for sure whether there had been any more in the first half before I started noticing their repeated appearance.

 

Anyway, assuming I "only" held five 4441 hands over the course of the 24 boards, I found myself wondering how unusual an occurence this was. According to my calculations, the chances of a given hand having a 4441 shape are just under 3%, so the expected number over the course of 24 boards is 0.72, but I wonder whether anyone can quickly work out the probability of at least 5 such hands. (I recall learning back in the dim and distant past - perhaps 35 years ago - about using a Normal approximation to a Binomial distribution, which might be one way of tackling this, but I've no real idea of how valid the approximation might be for these sort of numbers....)

 

Given your observation of 3 in the first 12 boards, the probability of getting exactly 5 in 24 boards is about 4.36% - a bit over one every 25 times that it happens.