As I understand it, independence of A and B can be informally established by asking whether learning something about one of those events tells you something new about the other. This must be borne out mathematically, however. For example:
If P(A|B) = P(A), then A and B are independent.
And if P(A & B) equals P(A) x P(B), then A and B are independent.
The above imply that P(B|A) = P(B)
It’s this last statement that confuses me, at least in application to certain cases. For example:
You devise a way to randomly choose a number from all real numbers, uniformly distributed. The probability of the chosen number being prime is 0, given that 0% of the reals are prime. Similarly, choosing the number 2 from the set of all real numbers has a probability of 0. Likewise, the probability of choosing a 2 from the set of all prime numbers has a probability of 0. Given that 2 is a prime number, it seems that choosing a 2 and choosing a prime number must be dependent events, at least in one direction (namely, if I know I’ve chosen a 2, then I’m certain I’ve chosen a prime number). Here’s what I mean:
P(2|prime number) = P(2) = 0 (passes for independence)
P(prime number|2) = 1 (i.e., not 0, or P(prime number), and so fails for independence)
But can also test as follows:
P(2 & prime number) = P(2) x P(prime number) = 0 P(2 & prime number) = P(2) x P(prime number|2) = 0 P(2 & prime number) = p(prime number) x P(2|prime number) = 0
Everything here comes out to 0, as I suppose it should. This also aligns with my understanding that anything with probability 0 is independent from any other event. (Right?) And yet, I’m stuck with the intuition that:
If I learn I got a 2, I know I got a prime number, wherein learning I got a prime is insufficient for updating my beliefs about getting a 2 (provided I really do believe that the probability of pulling a 2 from the primes is 0), and the same goes for learning I got an even, a natural, an integer, and so on. Yet I learn I got all those things if I learn I got a 2.
I’ve thought of other examples, though all of them deal with some single event occurring out of a set of infinite possible outcomes. E.g., pulling from the natural numbers: P(2|even) = P(2) = 0; but P(even|2) = 1 (rather than the P(even) = 1/2). So I imagine there’s something I’m naive about in the domain of infinite possible outcomes.
What am I missing?