Drawing Randomly from an Infinite Set

[Bork]

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?

[Vincent Lepore]

yes. infinite means just that.

vince

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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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He's "not even wrong". The fact is that "drawing randomly from the set of all natural numbers" is, at best, ill-defined. It's a bit like claiming, for exampe, that 2 is the only number that cares. (Your reaction should be WTF does it mean that a number 'cares'.)

If you look at the limit, as n goes to infinity of the density of prime numbers on the interval [1,n] then it tends to 1/(ln n) so it goes to zero, and the density of odd numbers tends towards .5.

[RiverTheNuts]

Lim such that x is prime = 0
x -> inf.

Lim such that x is odd = .5
x -> inf.

I think he's a moron, but what do I know

[UATrewqaz]

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

There are "infinity" of both.

I go with "no" cause some infinities are infinitly bigger that others

Ask your math prof. Your philosophy prof. is a moron who doesn't understand the concept of infinity.

To make this concept clearer, put it in physical terms. Take the Earth's atmosphere, which is composed of 78% nitrogen, 21% Oxygen, 1% Argon. Now if you go anywhere in the atmosphere and sample one molecule, you have a 78% percent chance of it being nitrogen, 1% of it being Argon. If you double the size of the atmosphere, same thing. Expand Earth's atmosphere to fill the entire universe, the ratios don't change. If you make the universe infinite, go to any spot and sample a molecule, the ratios still don't change, Argon is always far less likely to be picked.

In the prime number example, the incidence of prime numbers actually decreases as the size of number set increases. So the chance of picking a prime number goes to 0 as N goes to infinity. Note that it never actually reaches 0. The chance of picking an odd number stays the same at 0.5 no matter how large N gets.

You can tell your professor he's a [censored] monkey.

edit: Looking at this I find it hard to believe anyone can be so retarded. I'm guessing this was a homework question and I just did it for you /images/graemlins/smile.gif

[BruceZ]

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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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It is impossible to "draw randomly" from an infinite set of numbers, if "randomly" means that each number has the same probability (uniform distribution). This is because the sum of the probabilities of all the numbers must equal 1, and there is no probability p that we can assign each number such that the sum of an infinite number of these probabilities is equal to 1. That is, p would have to be smaller than any positive real number, or else the sum of the probabilities would diverge to infinity. It could not be zero, because then the infinite sum of zeros would be zero, not 1 (by definition since the limit of partial sums is zero).

Of course we are welcome to use some probability distribution which makes some numbers more likely than others, so that the sum of the probabilities converges to 1, even if there are still infinitely many possible numbers.

The best we can do for equal probabilities is to consider a uniform distribution from 1 to +N, where we let N become arbitrarily large. That is, all integers from 1 to +N can be chosen with equal probability 1/N, while integers outside this range have probability 0. Then we can answer your questions in the limit as N goes to infinity. Note however that we will always be considering a finite number N, but we allow N to be arbitrarily large.

Under these conditions, the number of odd numbers will always be N/2 for even N, and (N+1)/2 for odd N, so the probability of drawing an odd number will be 1/2 as N -> infinity. On the other hand, the fraction of prime numbers will decrease as N -> infinity, and the probability of drawing a prime number will go to 0 as N -> infinity.

[Jonas Wa]

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It is impossible to "draw randomly" from an infinite set of numbers, if "randomly" means that each number has the same probability (uniform distribution). This is because the sum of the probabilities of all the numbers must equal 1, and there is no probability p that we can assign each number such that the sum of an infinite number of these probabilities is equal to 1. That is, p would have to be smaller than any positive real number, or else the sum of the probabilities would diverge to infinity. It could not be zero, because then the infinite sum of zeros would be zero, not 1 (by definition since the limit of partial sums is zero).


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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>0.5)=0.5.

[Bork]

Nope, not a homework question.

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.

You guys propose that there are more general ways to calculate probabilities but you havent shown why his inference is flawed. Here is his inference: things with 1:1 correspondence within a set are equally likely to be drawn from that set.

Simply saying there are different sized infinites doesn't work because you can take every unique odd number and find a unique prime to assign to it. They are still clearly in one to one correspondence, different sized infinites or not.

[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

[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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It is a test..to see who blindly accept statements from their "superiors" and who will exhibit an enquiring mind to question authority and show independence of thought. and you have passed..let him/her know!