Probability for an infinite set

Question

The way probability is defined as the expected value works for finite sets. The probability of getting heads is out of two possible outcomes, heads or tails. If we asked the probability out of an infinite set, like the set of positive integers, we could take a limit. For example, as this site discusses, if we ask "What is the probability of a random integer being divisible by 5?", we can still answer that question, and the answer if 1/5.

However, if we ask the question of what is the probability that of a random integer is 5, and applied the same process, we would get a limit of zero.

Is it possible to ask what is the probability that a random number is 5 in a way that makes sense mathematically?

Answer 1

I think you have to define rather precisely what your random process is for choosing an integer. For example this process allows any non-negative integer to be chosen but gives each one a finite probability:

1. Randomly choose a digit from 0 to 9.
2. Toss a coin. If you get heads, go to step 1.
3. When you finally get tails, string together the chosen digits to make your random integer.

The process is random and obviously able to produce any positive integer or zero, but it's also heavily weighted towards lower numbers. In fact single-digit integers will occur more than half the time (since we might get a string of digits of which all but the last are zero).