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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.