16 Oct 10
Originally posted by twhiteheadHis example is a bit weird since people are not random number generators, but in essence any measurement of a continuous variable will give a probability zero result.
I would counter that probability also depends on the concept of randomness. So your example seems counter intuitive because the choices in question are not random. Whether this makes you claims wrong I am not certain, but you probably are.
16 Oct 10
Originally posted by twhiteheadI'd say that you haven't understood the example. Yes there are issues with but they are not what you point...
I would counter that probability also depends on the concept of randomness. So your example seems counter intuitive because the choices in question are not random. Whether this makes you claims wrong I am not certain, but you probably are.
But carrying on with your criticism: first off all please define randomness or at the very least just tell me if your using this technical term with its technical meaning and then explain to me what exactly do you mean by this: the choices in question are not random.
And this also Whether this makes you claims wrong I am not certain, but you probably are. Are you saying that the the probability of choosing a rational number on the real line isn't 0 and that the probability of using an irrational number on the real line isn't 1?
17 Oct 10
5 edits
Originally posted by adam warlockI tend to steer clear of stats and probability but since we're talkng about the real world and not an idealised mathematical one, don't we need to invoke conditional probabilty here?
I'd say that you haven't understood the example. Yes there are issues with but they are not what you point...
But carrying on with your criticism: first off all please define randomness or at the very least just tell me if your using this technical term with its technical meaning and then explain to me what exactly do you mean by this: the choices ...[text shortened]... isn't 0 and that the probability of using an irrational number on the real line isn't 1?
Ie: what is the probability that a person wll pick a rational number in [0,1] given some finite upper bound for the amount of time a person can spend trying to express any number (before they die, say!)
17 Oct 10
Originally posted by AgergIn that case wouldn't you have to do the same with all probability problems involving humans? Mathematics is always about idealizations because if we are truly interested in the real world you wouldn't solve any problem you'd just end up categorizing all the variables that might affect the outcome of the experience and these, in principle, are infinite.
I tend to steer clear of stats and probability but since we're talkng about the real world and not an idealised mathematical one, don't we need to invoke conditional probabilty here?
Ie: what is the probability that a person wll pick a rational number in [0,1] given some finite upper bound for the amount of time a person can spend trying to express any number (before they die, say!)
Th problem with the example I gave is more or less what KN said. The thing is that most people you'd encounter on the street are exactly fluent in the language of mathematics and would choose rational numbers because they are more used in dealing with rational numbers than with rational numbers.
For this problem to be unbiased you'd have to be biased on your sample: you'd have to choose people that are as used with dealing with rational numbers as they are used in dealing with irrational numbers: mathematicians,physicists and engineers come to mind. And even still I suspect more than 50% would choose rational numbers.
Anyway my example was just to illustrate a very counter-intuitive result of real analysis (measure theory if you prefer) even though the set of rational numbers and the set of irrational numbers have an infinite amount of elements in a very precise sense the set of irrational numbers have infinitely more elements than the set of rational numbers.
And all of this is totally independent of the the trivial result that any event in a sample space that is continuous has a 0 probability of happening and yet these events happen all the time. (Assuming, of course, that the probability density function is well behaved).
17 Oct 10
5 edits
Originally posted by adam warlockI don't (and didn't) disagree with the point you're making in principle, and if I continue this argument I'll only be muddying it with pedantry.
In that case wouldn't you have to do the same with all probability problems involving humans? Mathematics is always about idealizations because if we are truly interested in the real world you wouldn't solve any problem you'd just end up categorizing all the variables that might affect the outcome of the experience and these, in principle, are in ll the time. (Assuming, of course, that the probability density function is well behaved).
The only thing I was trying to get at is it's not so unreasonable to say that the probability of some human picking a specific number from the infinitely large set [0:1] is zero as it is to say that the probability they will choose some specific subset Qn[0,1] from the infinitely larger set [0,1] is zero. The latter seems overly extravagant with 'picky' details one would neglect in that most people (I assert) simply don't know how* to articulate any numbers that aren't in Q
*(I suppose most people are familiar with the number pi...but recognising it is irrational and taking, say, a quarter of it, or it's reciprocal??? that's a tall ask for most of the general populace)
17 Oct 10
Originally posted by AgergOk, but just tell me if this doesn't seem counter intuitive:
The only thing I was trying to get at is it's not so unreasonable to say that the probability of some human picking a specific number from the infinitely large set [0:1] is zero as it is to say that the probability they will choose some specific subset Qn[0,1] from the infinitely larger set [0,1] is zero.
Let X the set of all rational numbers in [0,1] and let Y be the set of all irrational numbers in [0,1].
What is the value of P(X) and P(Y)?
P(X)=0
P(Y)=1
17 Oct 10
Originally posted by adam warlockErrata:
In that case wouldn't you have to do the same with all probability problems involving humans? Mathematics is always about idealizations because if we are truly interested in the real world you wouldn't solve any problem you'd just end up categorizing all the variables that might affect the outcome of the experience and these, in principle, are in ...[text shortened]... ll the time. (Assuming, of course, that the probability density function is well behaved).
The problem with the example I gave is more or less what KN said. The thing is that most people you'd encounter on the street aren't exactly fluent in the language of mathematics and would choose rational numbers because they are more used to dealing with rational numbers than with irrational numbers.
17 Oct 10
Originally posted by adam warlockBy 'random' I meant that the probability of of selecting any given number in the set is equal.
But carrying on with your criticism: first off all please define randomness or at the very least just tell me if your using this technical term with its technical meaning and then explain to me what exactly do you mean by this: the choices in question are not random.
Your probability calculation assumes that the probability of selecting any given number in the set is equal. This is simply not true.
If you have a weighted die that is more likely to fall on a 6 in real life, it is wrong to claim that the probability of it falling on a 6 is 1 in 6 because it has six possible out comes.
And this also Whether this makes you claims wrong I am not certain, but you probably are. Are you saying that the the probability of choosing a rational number on the real line isn't 0 and that the probability of using an irrational number on the real line isn't 1?
I am saying that the probability of a randomly chosen human being choosing a rational number on the real line is very close to one - and quite definitely not zero.
17 Oct 10
2 edits
Originally posted by twhitehead
By 'random' I meant that the probability of of selecting any given number in the set is equal.
Your probability calculation assumes that the probability of selecting any given number in the set is equal. This is simply not true.
If you have a weighted die that is more likely to fall on a 6 in real life, it is wrong to claim that the probability of ing a rational number on the real line is very close to one - and quite definitely not zero.
By 'random' I meant that the probability of of selecting any given number in the set is equal.
Very poor definition of random. By that definition all sample spaces that don't have equiprobable events aren't representing random experiments. This is clearly false.
Your probability calculation assumes that the probability of selecting any given number in the set is equal
No it doesn't. And I'd like to know where do you get this idea from. All it assumes are very standard results of real analysis (measure theory).
I am saying that the probability of a randomly chosen human being choosing a rational number on the real line is very close to one - and quite definitely not zero.
That's what I (kinda) said too.
I already explained what is the "fault" on my example and I think it'd be good for you to read it.
19 Oct 10
2 edits
Originally posted by adam warlock'Wrong answer' I suspect but I'd have to say no in this case. The rationals I just picture as "small" punctures on the real line.
Ok, but just tell me if this doesn't seem counter intuitive:
Let X the [b]set of all rational numbers in [0,1] and let Y be the set of all irrational numbers in [0,1].
What is the value of P(X) and P(Y)?
P(X)=0
P(Y)=1[/b]
More counter intuitive to me is the fact that the set of even numbers is equal in size to the set of integers...I understand why it's true of course, but it still betrays my intuition.
19 Oct 10
Originally posted by adam warlockI intended it to be a definition of what I meant by random in that instance, not in every usage of the word.
Very poor definition of random. By that definition all sample spaces that don't have equiprobable events aren't representing random experiments. This is clearly false.
I do realize that you may get the same result for other types of randomness, but I still maintain that your calculation assumes my definition.
No it doesn't. And I'd like to know where do you get this idea from. All it assumes are very standard results of real analysis (measure theory).
I am afraid I haven't studied measure theory or real analysis. Could you explain what this standard result is and what it does assume?
Clearly it does assume that a highly biased selection (such as people) is being used, so there are assumptions. How does it define those assumptions?
I already explained what is the "fault" on my example and I think it'd be good for you to read it.
I don't understand your explanation.
19 Oct 10
1 edit
Originally posted by twhiteheadI don't think that going for more technical definition(s) is helpful as the discussion would become too abstract and solve little as its definition depends whether you're a frequentist or bayesian.
Clearly it does assume that a highly biased selection (such as people) is being used, so there are assumptions. How does it define those assumptions?
You're correct in that one has to specify a distribution, but adam's point was that any distribution over a continuous support will give you individual draws of probability zero.
And we're talking about very standard distributions here like the normal (bell curve), exponential, logistic, uniform over an interval, etc.
I would suspect that what you have in mind is not a different definition of random, just a different distribution. For example, if you ask people to "simulate" long sequences random coin tosses and compare it to real coin tosses then it is possible to identify which are the real ones and which ones are fake. Is this a different definition of randomness? To some purists it might be but if you see it as coming from the sampling then even a purist might agree that it's the same notion.
19 Oct 10
1 edit
Originally posted by AgergEven more counter-intuitive, then, will be fact that the set of rational numbers is equal in size to the set of integers!
'Wrong answer' I suspect but I'd have to say no in this case. The rationals I just picture as "small" punctures on the real line.
More counter intuitive to me is the fact that the set of even numbers is equal in size to the set of integers...I understand why it's true of course, but it still betrays my intuition.
