[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.