Drawing Randomly from an Infinite Set
 

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?

Re: Drawing Randomly from an Infinite Set
 

yes. infinite means just that.

vince

Re: Drawing Randomly from an Infinite Set
 

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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.

Re: Drawing Randomly from an Infinite Set
 

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

Re: Drawing Randomly from an Infinite Set
 

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

There are "infinity" of both.

Re: Drawing Randomly from an Infinite Set
 

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

Re: Drawing Randomly from an Infinite Set
 

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 [img]/images/graemlins/smile.gif[/img]

Re: Drawing Randomly from an Infinite Set
 

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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.

Re: Drawing Randomly from an Infinite Set
 

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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.

Re: Drawing Randomly from an Infinite Set
 

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.