Higher than "chance"

[drkitten]

The probability is not zero but reciprocal infinity:
This link explains it as understand it:

The link is utterly, utterly, wrong. He's gotten the formalisms so thoroughly wedged it's hardly worth trying to take them out.

Examples :

Infinity is not a real number. It is a concept. Normal algebraic rules often cannot be applied to the concept of infinity.

This is correct.

It is possible to divide by zero. Any non-zero number divided by zero is infinity.

This is incorrect

It is possible to divide by infinity. Any non-zero number divided by infinity is the reciprocal of infinity (1 / infinity), NOT ZERO!

Not only is this incorrect, but taken in conjunction with the previous quotation "proves" that division is not the inverse of multiplication, which throws essentially all of number theory out the window.

Zero divided by zero is zero, NOT undefined/indeterminate.

This is incorrect.

Zero divided by infinity is zero

This is incorrect.

You can multiply anything by zero, or multiply zero by anything, and the result is nothing. If you multiply nothing by infinity, the result is nothing, NOT undefined/indeterminate.

This is incorrect.

In the same way that anything minus the same number is zero, infinity minus infinity is zero.

This is not only incorrect, perniciously so. First of all, any six-year old can see that if you take all the even numbers away from all the odd numbers, you still have something left. But more perniciously, as the author later points out :

Note that if infinity minus infinity is zero, then associativity does not apply.

... which means that addition is no longer a group (let alone a field), and all of algebra flies out the window.

In the same way that anything divided by the same number is 1, infinity divided by infinity is 1

Again, obviously wrong to any six-year old.


Ashles, I certainly hope that wasn't your web site -- but the "mathematics" presented there not only aren't standard (which by itself is bad enough), but they're not even self-consistent. The author of that page is working hard on becoming the Interesting Ian of mathematics.

 

[Ashles]

The link is utterly, utterly, wrong. He's gotten the formalisms so thoroughly wedged it's hardly worth trying to take them out.

I didn't read through all of the page and now am wishing I had. His stuff about zero in particular is obviously incorrect.

I read the section about infinity being a concept and I thought it was correct in that anything divided by infinity would not be zero as you are
a) applying algebraic rules to a concept and
b) it would have a theoretically infinitely small non-zero value. Its value would be infinitessimal. (although I am beginning to think this second point doesn't apply simply because it is still trying to put infinity, albeit theoretically, in an equation, and this isn't meaningful)

So why he then goes on to break the 'infinity is a concept' rule and happily shove it in all sorts of equations is rather weird.

Again, my bad for not reading the whole page.

But surely it is still fair to say anything divided by infinity is not zero as it is not a meaningful expression in the first place. Infinity just cannot be divided, added, factored etc. as it is not a number.

This appears to be a better link.

 

[drkitten]

But surely it is still fair to say anything divided by infinity is not zero as it is not a meaningful expression in the first place. Infinity just cannot be divided, added, factored etc. as it is not a number.

Yes, but when you're talking about the probability of a point-event in a continuous probability space, you're not simply "dividing by infinity." Probabilities in continuous distributions are defined as integrals over the space, and of course, the integral of any function from a to a is exactly zero.

But if you don't like that argument, there's another, more constructive proof available. Probabilities are defined as real numbers in the range [0,1]. If you assume a non-zero probability epsilon for a point event in any continuous distribution, I can easily construct an event interval with probability epsilon/2 with the point event as a subset. This is a contradiction, therefore the probability of point event in a continuous distribution is exactly zero. (q.e.d.)

 

[Beth]

Well that's not zero though. We could round 0.5 probability to 1, but it wouldn't make heads coming up on a coin toss a certainty.[/q]

You're correct that it's not truly non-zero. However, it's not at all uncommon for extremely low probabilities to be rounded to zero, particularly when discussing the matter in lay terminology.

How close to zero before you round? Well, that's up to you. Few people would round a probability of 0.5 to 1.0, particularly since all probabilities lie between 0 and 1, it's a huge relative shift. But most people would go ahead and declare a probability of 0.99999999999999 as being a certainty and a probability 0.0000000000000001 as zero probability. They are close enough for almost all practical applications. Shucks, why Randi's willing to accept a probability lower than only 0.000001 to win the$1,000,000 challenge. That's not zero either, but close enough.

The probability is not zero but reciprocal infinity:
This link explains it as understand it:

What we have to remember is that infinity is a concept rather than an actual number.

new drkitten responded to this far better than I could.

Beth

 

[CurtC]

Any non-zero number divided by infinity is the reciprocal of infinity (1 / infinity), NOT ZERO! Reciprocal infinity is as close as you can get to zero without actually being zero.

So I guess 0.999... is as close as you can get to one without actually being one. [insert roll-eyes smiley here, I can't remember how to code it]

 

[CurtC]

However, it's not at all uncommon for extremely low probabilities to be rounded to zero, particularly when discussing the matter in lay terminology.

Really!?! I have never heard anyone say, even in the lay world, that the probability of some very unlikely, but still possible, event is zero. Have you?

 

[rppa]

Content deleted: Duplicate post.

As others have probably noted, the database server is being extremely flaky right now.

 

[rppa]

Originally posted by Ashles
This reminds me of a conversation I had ages ago with a co-worker.
We were talking about the lottery and he confidently declared some combinations of numbers would never come up, for example 1,2,3,4,5,6

Almost definitely it won't.

Bad news for your theories of how things work in real life, Ian:

http://www.clublottery.co.za/news/lotto-history.htm

History was made this Saturday, when a record 33 people shared the Jackpot prize. What was even more amazing was that the numbers chosen by the winners were a combination of sequential uneven numbers... Lotteries abroad have seen similar incidents. A record was set in Australia on July 24, 1993 when 150 people shared the A$2.3 million Jackpot. The numbers drawn followed a straight line on the entry coupon.

Of course they are equally likely. Why doesn't anyone ever say anything on this forum which I don't already understand?

So you agree that the chance of a deal with each player getting 13 cards of the same suit, is the same as any other specific set of four hands being dealt, for instance this one?

http://www.baronbarclay.com/sheinwold/123104.html

(nothing special, just a randomly selected bridge column from the newspaper with one particular deal)

Couldn't you say that the chance of this particular deal being dealt is so small that it just isn't, for all practical purposes, going to happen?

If not, what are the chances of this particular deal?

 

[drkitten]

Really!?! I have never heard anyone say, even in the lay world, that the probability of some very unlikely, but still possible, event is zero. Have you?

Does the web count?

See this page from Australia

In particular :

The idea that something may have a probability of zero of occurring but is still possible, is often used in mathematics. Suppose you have a new whiz-bang idea that is correct in all cases except for a few minor ones. If you can show that you have zero probability of encountering these minor cases, then this is just as good as showing it is true for ALL possible cases.

This isn't even the lay world.... this is a math department web site, although it's obviously written to the lay audience..

 

[Timothy]

Does the web count?

See this page from Australia

This isn't even the lay world.... this is a math department web site, although it's obviously written to the lay audience..

If the web counted, then all the drivel on here would count as well.

And it's not a math department website, it's the personal web page of a student in the math department.

 

[Timothy]

One must be very careful in distinguishing between informal and formal language when dealing with mathermatical issues.

The "rounding" argument of zero probablilty is false. The fallacy lies in using the same word to mean "zero" and "close enough to zero not to make any appreciable difference".

In this case "close enough to zero not to make any appreciable difference" must mean close enough that in the realm of all possibilities, not even a single one exists. Therefore it must be the same as "zero" or else the argument falls apart, as we've seen.

- Timothy

 

[Interesting Ian]

You know, if we are not allowed to actually delete our posts, why is that option even shown to us?


I began responding to posts in this thread, but every single post Interesting Ian made was so outrageously more silly than the one before that I had to stop trying. The combination of unwilling to understand and unable to understand cannot be overcome.

Could you let me know what I have said that you believe to be incorrect? If I'm convinced by your explanation, I'll readily admit it. I don't expect you to respond to this post though

 

[drkitten]

If the web counted, then all the drivel on here would count as well.

Ahem.

CurtC claimed that he had never seen anyone make the claim.

There's someone making the claim.

And there's a very big difference between Mr. Duong's well-written page and the drivel here -- Mr. Duong is a genuine expert in his field and his writings are correct.

In this case "close enough to zero not to make any appreciable difference" must mean close enough that in the realm of all possibilities, not even a single one exists. Therefore it must be the same as "zero" or else the argument falls apart, as we've seen.

This is simply false. For example, if you select a real number with uniform probability (the bounding interval doesn't matter), the probability of getting a rational number is zero. (This is easily proven as the rational numbers have measure zero on the real line.) However, rational numbers rather obviously exist -- and in fact, there are infinitely many of them. There are simply enough numbers that aren't rational that the probability of selecting one is zero.

 

[Timothy]

Please read my post with more care.

I was not suggesting that the argument about zero probability events occuring was wrong, I was stating that Beth's argument about being close enough to zero that rounding to exactly zero was facetious.

- Timothy

 

[Interesting Ian]

Also the probability of any particular value occurring in a continous distribution (such as the normal distribution) is zero. This has to do with the fact that there are an infinite number of values possible, thus the probability of any one particular value occurring is zero.

I think what Beth says here is what I have in mind. I can't be sure because I do not understand a lot of the terminology she employs (I stopped doing mathematics at the age of 16). But let me put it in the way I would explain it so that you can understand (well, you probably won't be able to, but I'll give it a shot! )

Imagine a computer generating a genuine random number between 0 and 1. Imagine the number of digits after the decimal point go on forever -- similar to pi or square root of 2 etc (and ignore the fact that it is actually impossible for a computer to do this). Thus if 0-1 represents a length, choosing such a number will specify a point between these 2 values with infinite precision. Crucially though, an infinite number of differing numbers are available to be chosen which all fall into the range.

Now, since there are an infinite number of such numbers, if you specify such a particular number before the computer generates this number randomly (of course you can't in reality specify a number since it is infinitely long, but ignore this point), then there will be 1/infinity of the computer choosing that number.

1/infinity is P=0

However

Whatever number the computer chooses will be a specific number. And even if beforehand there is a zero probability of it being generated, it was generated nevertheless!.

Basically if you think of such a specific number, then it cannot be generated by the computer. This is so despite the fact that the computer does generate a number which had zero probability of occurring

I explain all this in the other thread once it was explained to me that logically impossible does not equate to zero probability. I understood what they were saying, but then they couldn't understand that zero probability does mean impossible, even though not logically impossible. LMAO

Another interesting point to make is that despite the fact that the probability of the computer generating any specifiable single number is zero, we know that the computer does actually choose one number. This means that in this instance adding up an infinite number of zero probabilities actually equals 1!

 

[Dr Adequate]

I agree it is far to unlikely to have actually occurred, therefore I am not simply the product of a sperm and an egg.

But I've already argued about this in another thread about a year ago, and no-one could understand my argument.

Oh... my... word...

 

[Dr Adequate]

I explain all this in the other thread once it was explained to me that logically impossible does not equate to zero probability. I understood what they were saying, but then they couldn't understand that zero probability does mean impossible...

I guess the reason that they "couldn't understand" this is that it's rubbish.

 

[Interesting Ian]

quote:
--------------------------------------------------------------------------------
It is possible to divide by zero. Any non-zero number divided by zero is infinity.
--------------------------------------------------------------------------------


This is incorrect


quote:
--------------------------------------------------------------------------------
It is possible to divide by infinity. Any non-zero number divided by infinity is the reciprocal of infinity (1 / infinity), NOT ZERO!
--------------------------------------------------------------------------------



Not only is this incorrect, but taken in conjunction with the previous quotation "proves" that division is not the inverse of multiplication, which throws essentially all of number theory out the window.


quote:
--------------------------------------------------------------------------------
Zero divided by zero is zero, NOT undefined/indeterminate.
--------------------------------------------------------------------------------



This is incorrect.


quote:
--------------------------------------------------------------------------------
Zero divided by infinity is zero
--------------------------------------------------------------------------------



This is incorrect.


quote:
--------------------------------------------------------------------------------
You can multiply anything by zero, or multiply zero by anything, and the result is nothing. If you multiply nothing by infinity, the result is nothing, NOT undefined/indeterminate.
--------------------------------------------------------------------------------



This is incorrect.


quote:
--------------------------------------------------------------------------------

In the same way that anything minus the same number is zero, infinity minus infinity is zero.
--------------------------------------------------------------------------------



This is not only incorrect, perniciously so. First of all, any six-year old can see that if you take all the even numbers away from all the odd numbers, you still have something left. But more perniciously, as the author later points out :


quote:
--------------------------------------------------------------------------------

Note that if infinity minus infinity is zero, then associativity does not apply.
--------------------------------------------------------------------------------



... which means that addition is no longer a group (let alone a field), and all of algebra flies out the window.


quote:
--------------------------------------------------------------------------------
In the same way that anything divided by the same number is 1, infinity divided by infinity is 1
--------------------------------------------------------------------------------



Again, obviously wrong to any six-year old.


Ashles, I certainly hope that wasn't your web site -- but the "mathematics" presented there not only aren't standard (which by itself is bad enough), but they're not even self-consistent. The author of that page is working hard on becoming the Interesting Ian of mathematics.

Excuse me??? What the hell are you talking about?? Where have I ever said anything that is incorrect about mathematics??

Please give evidence of anything I have said regarding mathematics that is incorrect!!!.

I actually agree with most of what you said regarding that person's website. I say most because I disagree with you where the author says:

--------------------------------------------------------------------------------
It is possible to divide by zero. Any non-zero number divided by zero is infinity.
--------------------------------------------------------------------------------

And you comment:

This is incorrect.

It does not seem to be incorrect to me, but I shall avoid making definitive statements since Beth says she agrees with everything you say and I respect her judgement somewhat more than your facile proclamations

Wow, just really wish I was a student in one of your classes. I'd really love to take the p*ss out of you.

Moron.

 

[Interesting Ian]

Originally posted by Interesting Ian
I explain all this in the other thread once it was explained to me that logically impossible does not equate to zero probability. I understood what they were saying, but then they couldn't understand that zero probability does mean impossible...
--------------------------------------------------------------------------------

I guess the reason that they "couldn't understand" this is that it's rubbish.

Another moron calling himself a Dr

So, it's possible that an event which has zero probability of occurring to occur anyway?

You really want to maintain this? Or are you simply confused between something being possible and something being logically possible??

 

[Dr Adequate]

Er... Ian... calling dr k a "moron" won't stop every single detail of his post from being absolutely correct. It just makes you look like a stupid petulant child.