Probability question.

Originally posted by Shallow Blue
I call bunkum. Doesn't matter whether on your part or on the part of the prof who signed your B.Math. If there was no probability course in it, you're not a B.Math., however much you may have a paper saying you are.
FFS, we did probability in high school!

Richard

Nonsense! Here at manchester with a combined maths with scientific computing degree (2/3, 1/3 split respectively) there is no requirement to take any probability.

Originally posted by adam warlock
The fact that I claimed virtually the opposite of that hasn't still hit you in the head? Do you really think that I claimed that or are you just taking the pi$$?

I missed that. I think you were very unclear in when you said it. I'll put it down to your different definition of 'probability'. You seem to have assumed that because there is a technical mathematical use for the term then everyone else must use that definition or else.

I think what confused me is that you several times suggested that probability zero event do happen. Now you are denying it (for this example).

Originally posted by twhitehead
I missed that. I think you were very unclear in when you said it. I'll put it down to your different definition of 'probability'. You seem to have assumed that because there is a technical mathematical use for the term then everyone else must use that definition or else.

I think what confused me is that you several times suggested that probability zero event do happen. Now you are denying it (for this example).

I missed that. I think you were very unclear in when you said it.

Yes I was. I only said it two or three times and even went as far as putting on bold one time. Pretty unclear if you ask me.

You seem to have assumed that because there is a technical mathematical use for the term then everyone else must use that definition or else.

Silly me! Just because we're on the science forum talking about a scientific issue I assumed that we were supposed to use paradigmatic scientific definitions.
But of course now I know better: we can all use our own half-baked definition of scientific terms on scientific discussions. I'm sure that will help a lot on the job of exchange ideas...

I think what confused me is that you several times suggested that probability zero event do happen.

I didn't suggest that. I affirmed that because I know it is true. And everyone that knows what he's talking about when talking about Probability and Statistics also knows that to be true.

Just to be clear on this. What truth value do you assign to the following sentence: 0 probability events never happen.

Now you are denying it (for this example).

No I'm not. And once again I'd like to know where are you taking this from. A single of quote of me denying it will suffice.

I really do hope that you have a very strange kind of humor and are taking the pi$$.
On the other hand if you are being serious with all of this I'll have to put you on the conceited idiot category and advise you to know your stuff before going and running your mouth on topics you apparently know very little about.

Originally posted by twhitehead
I missed that. I think you were very unclear in when you said it. I'll put it down to your different definition of 'probability'. You seem to have assumed that because there is a technical mathematical use for the term then everyone else must use that definition or else.

I think what confused me is that you several times suggested that probability zero event do happen. Now you are denying it (for this example).

I think you are mixing the concepts of probability with probability distribution. What you have when you ask people around is just draws from different probability distribution than the uniform (which is the one where every draw has the same probability). This is not a different definition of probability.

Also, that many people confuse both is not a reason to conform to that and accept their confusion. Instead of insisting, perhaps you should try to learn understand the difference and move the discussion forward. If this was about an "everyman's" view of evolution I bet you wouldn't be accepting some muddled and imprecise view about what it is.

Originally posted by Palynka
I think you are mixing the concepts of probability with probability distribution. What you have when you ask people around is just draws from different probability distribution than the uniform (which is the one where every draw has the same probability). This is not a different definition of probability.

Well perhaps you can explain it in relation to the given example.
adam warlock seems intent on telling me I am wrong, but not really explaining why. He seems more interested in pointing out my ignorance on the subject (which I do admit).

This is my understanding:
If I asked "what is the probability of randomly picking a rational from the set of reals?" then my answer would be zero.
But if I asked what is the probability of a human being picking a rational from the set of reals my answer would be 'close to one'.

My problem is with the sentence:
"What is the probability of choosing a rational number on the real line?".
It doesn't state who is choosing.
adam warlock suggests that the answer is zero even though the chooser is not specified.

Originally posted by twhitehead
Well perhaps you can explain it in relation to the given example.
adam warlock seems intent on telling me I am wrong, but not really explaining why. He seems more interested in pointing out my ignorance on the subject (which I do admit).

This is my understanding:
If I asked "what is the probability of randomly picking a rational from the set of reals
adam warlock suggests that the answer is zero even though the chooser is not specified.

Yes, that sentence is incorrect/imprecise because, first, he doesn't specify a distribution (if you want me to define what I mean by distribution please say so) although for people used to probability when you don't specify a distribution it usually means the uniform (i.e. every point has the same probability of being drawn). But you're right that a distribution should be specified. Secondly (and that's more a pedantic technical issue) you cannot draw from a uniform over the real line because that distribution doesn't exist.

But what he meant was something like: "What is the probability of drawing a rational number from the interval [0,1], if the probability of drawing any given real is the same across the whole interval?"

And here the probability of drawing a rational number is zero because there are infinitely more reals than rationals (imprecise technically, but that's the idea).

Who draws? Usually we say that nature draws it because we want it to be a purely random process. In this case, it doesn't matter who or what draws because we specified the distribution already. All we want is that the "drawer" conforms to the condition that every real is equally likely.

Originally posted by twhitehead
My problem is with the sentence:
"What is the probability of choosing a rational number on the real line?".
It doesn't state who is choosing.
adam warlock suggests that the answer is zero even though the chooser is not specified.

There's no chooser. Probability isn't a fraction between favorable cases and totality of cases. Probability is a function and is calculated by evaluating an integral.

This statement is just a rephrasing of Cantor's diagonal argument. And I had already told you that. If you don't know what Cantor diagonal argument is about either you google it or you ask for a clarification.

Originally posted by Palynka
Yes, that sentence is incorrect/imprecise because, first, he doesn't specify a distribution

I don't need to specify a distribution. This is standard text book measure theory basic knowledge.

The set of reals has 0 measure hence 0 probability. Don't believe me just ask any mathematician you know or consult the relevant bibliography.

Originally posted by adam warlock
I don't need to specify a distribution. This is standard text book measure theory basic knowledge.

The set of reals has 0 measure hence 0 probability. Don't believe me just ask any mathematician you know or consult the relevant bibliography.

Distributions over such sets can have mass points. I know it's common to omit this and pendantic to insist but if asked one should then make it clear.

Edit - Especially since the example of "people choosing" will clearly have mass points...

Edit 2 - You meant the set of rationals has 0 measure, right?

Originally posted by Palynka
Distributions over such sets can have mass points. I know it's common to omit this and pendantic to insist but if asked one should then make it clear.

Edit - Especially since the example of "people choosing" will clearly have mass points...

Edit 2 - You meant the set of rationals has 0 measure, right?

Yes, I meant the set of rationals.

The example of people choosing was to juxtapose a mathematical result with a real world example.

Using only probability methods it is impossible to show that P(X)=0, X being the set of all rational numbers, and one has to use measure theory to establish that fact. That's why I said that no density distribution is needed here.
I don't even recall anyone talking about probability density functions when expressing this (in my view) very counter-intuitive result...

Originally posted by Agerg
Nonsense! Here at manchester with a combined maths with scientific computing degree (2/3, 1/3 split respectively) there is no requirement to take any probability.

All that means is that one should never hire a programmer (meaning in this case, of course, Mathematica prodder) who graduated at Manchester Polytechnic.

Seriously, though, probability is even more important in scientific computing than in the fundamental side. Leaving it out is a bad, bad choice, and shouldn't be allowed.

Richard

Originally posted by adam warlock
Yes, I meant the set of rationals.

The example of people choosing was to juxtapose a mathematical result with a real world example.

Using only probability methods it is impossible to show that P(X)=0, X being the set of all rational numbers, and one has to use measure theory to establish that fact. That's why I said that no density distribution is ...[text shortened]... obability density functions when expressing this (in my view) very counter-intuitive result...

It's possible to show that the set of rationals has Lebesgue measure 0 on the real line, but this is not a probability measure. So you can't actually show that P(X) = 0. There simply is no uniform over the real line. Anyway, we digress.

Originally posted by Shallow Blue
Seriously, though, probability is even more important in scientific computing than in the fundamental side. Leaving it out is a bad, bad choice, and shouldn't be allowed.

You think so? My maths degree had a fair bit of probability. I don't think I've ever actually used any of it as a software developer (and that included some scientific computing). In some applications, sure, but not in general.

On the other hand, I think probability is one of the most important areas of maths for the general public to understand. If I was asked "what is the point of learning maths", then a lot of the best examples are in probability. It should be compulsory, for example, for all lawyers and journalists. You can't properly understand evidence without it.

(In my opinion, obviously)

Originally posted by Palynka
It's possible to show that the set of rationals has Lebesgue measure 0 on the real line, but this is not a probability measure. So you can't actually show that P(X) = 0. There simply is no uniform over the real line. Anyway, we digress.

But using measure theory and probability you can show that P(X)=0 because with measure theories you don't need to have a probability density function in order to calculate probabilities.
For instance let us think about the cantor distribution.

Sometimes digressions are fun. In this case it was fun with you. *wink wink nudge nudge*

Originally posted by adam warlock
But using measure theory and probability you can show that P(X)=0 because with measure theories you don't need to have a probability density function in order to calculate probabilities.

Sure, but you need to use probability measure if you want to talk about probability. Only some measures are probability measures and the Lebesgue measure over the real line is not one of them. Measure 0? Yes. P(X)=0? No. There simply is no uniform over the real line.

Interestingly, yet another problem where there is a discontinuity in unbounded sets. The limit with bounded sets (interval over the real line) as you increase the set bound seems well-defined but is not the correct answer when you consider the unbounded set (the whole real line).