Drawing Randomly from an Infinite Set

[Vincent Lepore]

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They are still clearly in one to one correspondence, different sized infinites or not.

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The problem is that Philosophers take things literally while mathematicians know better.

Vince

He is technically correct because of how mathematics defines infinite sets. The set of all multiples of 10 is the same size as the set of naturals. He would be wrong if he used the set of all reals instead of naturals tho, since there are an infinite # of reals that can be assigned to each natural.

[Siegmund]

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He stills says that the odd numbers and prime numbers can be put in one to one correspondence, and thinks it follows from this that they are equally likely to be drawn.


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As many others have said, the first statement is correct; they CAN be put in 1-1 correspondence.

There exist measures in which the measure of the set of all odd numbers and the set of all even numbers are equal. BUT, in any such measure, either the measure of the odds (and some sets larger than the odds) will be zero [with one exception, see below], or the measure of the primes (and some sets smaller than the primes) will be infinite.

A probability measure is a measure for which m(Everything)=1. So the counting measure, under which m(primes)=m(odds)=infinity, isn't going to help us.

The definition of measure demands that the the measure of a union of disjoint sets is the sum of the measures of the sets. So we must have

m(primes) + m(nonprime odds) = m(odds) + m(2).

If m(primes)=m(odds), then m(nonprime odds)=m(2). This requires either a very unusual measure where a particular singleton weighs as much as an infinite set, or requires m(composite odds) to be zero.

I invite your professor to explain how he will convince me that a sampling scheme which never selects an odd number, or which selects 2 more than twice as often as 1 or 3, will meet a reasonable person's sense of what "random sampling" means.

[Vincent Lepore]

Suppose his professor had said you are just as likely to draw an even number as you are a prime number from the set of natural numbers. Would he be wrong? Of course not. The professor has infinitiy on his side.

Vince

[pzhon]

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gintron: He is technically correct because of how mathematics defines infinite sets. The set of all multiples of 10 is the same size as the set of naturals.

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UATrewqaz: The answer is "yes" because technically there are not "more" non-prime than prime numbers in the set.

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This point that both of you made does not have the conclusion that you seem to think it does. It still does not make sense to say that you are equally likely to draw a prime number as you are to draw an odd number. Please read what rufus and BruceZ wrote.

[alThor]

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He stills says that the odd numbers and prime numbers can be put in one to one correspondence,

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Sure.

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and thinks it follows from this that they are equally likely to be drawn.

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This conclusion is wrong because it is not even well defined, like rufus originally said.

Ask him "what distribution he has in mind". For example, when trying to pick an odd number from the set of integers, ask him how much probability he puts on each integer in advance, e.g. the probability that "1" is picked. When he says "each integer is equally likely", insist he tell you the exact number. If he tries to give a positive number, point out that the sum of probabilities will be infinite, so that isn't legitimate. If he says "zero" point out that the sum of probabilities will be zero, which also doesn't work. Therefore, they can't be "equally likely".

That is the flaw.

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I like some of these examples, but while they probably show that he is wrong in his conclusion, they don't pick out the mistake in his reasoning.

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You've now heard the flaw multiple times: There's no uniform distribution on the set of all integers.

alThor

[elitegimp]

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He stills says that the odd numbers and prime numbers can be put in one to one correspondence, and thinks it follows from this that they are equally likely to be drawn.


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you can put the odd numbers in one-to-one correspondence with the natural numbers. Does this mean that if you draw a number, the odds of drawing an odd number is the same as the odds of drawing a number? (hint: the odds of drawing a number given that you drew a number is 1)

</font><blockquote><font class="small">Svar till:</font><hr />

It is possible to draw randomly from an infinite set. If you look at an atom (numbers) at the time it will have a probility of 0. But if you look at a group of atoms (number) they could have a probility greater then zero. Take for instance uniform measure on [0,1]. Their is infintly many real number but P(x&gt;0.5)=0.5.

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From an uncountable infinite set (like [0,1]), yes. But, from an countable infinite set (like the set of itnegers, which the original question was about):

The Kolmogorov probability axioms state that the probability of a union of countably many disjoint sets is the sum of the probabilities of those sets. In a countable infinite set the entire set is a countable union of all singletons. Thus the probability of the entire set is zero (or infinite if the single events have probability a&gt;0), which is a contradiction as the probability of the entire set must be one.


{This assumes using Kolmogorov's probability axiomatization, maybe probability can be defined in some other way to make uniform distributions in countable infinite sets possible, but I'm not familiar with any other definitions. }

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He stills says that the odd numbers and prime numbers can be put in one to one correspondence, and thinks it follows from this that they are equally likely to be drawn.

I like some of these examples, but while they probably show that he is wrong in his conclusion, they don't pick out the mistake in his reasoning.

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He's assuming that the cardinality of a set has something to do with the measure of a set:

Here's a (relatively simple) counterexample. Let's say we consider the interval [0,1]. Depending on what you like, we can consider the rationals, or the reals on that interval.

Now, for the usual notion of probability, the chance of the randomly chosen value being on [0,.5] is going to be .5.

However, the cardinality of [0,.5] and [0,1] are the same since division by two (or multiplying by two if you're going in the other direction) is an easy 1-1 correspondance.

As has already been pointed out, if the inference were correct, you would be equally likely to pick an odd number as *any natural number*. Moreover, from the same inference you get that any even number is equal in likelyhood to be chosen to any natural number. Since the probabilities are exclusive, this leads to the conclusion that you're twice as likely to pick a natural number as you are to pick a natural number when picking a natural number.

I said, in my inital response that neither the probabiltility of picking an odd number, nor the probability of picking a prime is well-defined, so, without more context he might as well be asking what you get if you divide the color blue by the concept of justice.

[Trantor]

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My philosophy prof said that if you draw randomly from the set of all natural numbers you are just as likely to draw a prime number as you are to draw an odd number. Is he right or wrong? Why?

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Maybe you could ask him for a reference to a proof of his statement. Maybe you will find philosophers aren't into proofs:0