How does one calculate probability when there are infinite possibilities?

Question

For example, if I picked a random integer, from the infinite amount of integers, what would be the probability of picking, say 1? And how can this be extended? Say I have x + y = 0, and I pick a random integer each for x and y, what is the probability of it being a true statement?

Please answer keeping in mind I haven't really taken any course in probability or statistics.

Answer 1

It depends on how you're picking them.

Here's one way of picking a random integer. Flip a coin repeatedly until it comes up tails, and then write down the number of times you got heads before getting tails. Next, flip the coin one last time; if it comes up heads, write a negative sign before your number.

If you use this method, then the only sequence of coin flips which results in the number 1 is "heads, tails, tails", so the probability of getting 1 here is 1/8.

But here's a question. What happens if you use a method that has an equal chance of giving any integer (a discrete uniform distribution)? One where each integer is as likely as each other one?

The answer is: there is no such method. And here's why.

If you had such a method of picking a random integer, then there would have to be a probability, $p$, such that the probability of picking any given integer is $p$. The probabilities have to add up to 1, so $p$ would have to be a number such that, when you add up $p$ once for each integer (infinitely many copies of $p$), you get 1. But there is no number which behaves like that. If you try using 0 for $p$, the sum will come out as 0. If you try using any number greater than 0 for $p$, then the summation will diverge to infinity.

Answer 2

For both your questions the probability is 0. The simple way of finding such probabilities is if you can separate the infinite possibilities into a finite number of sets, each of them containing infinite number of elements, but where you can easily write a relationship between them. Here is an example: what's the probability that a random integer is divisible by 3? I can create 3 sets, numbers of the form $3k$, $3k+1$, and $3k+2$, with $k$ any integer. For all $k$ values I have an element in each of those sets, so the probability will be 1/3

Answer 3

As other answers have noted, there isn't a good uniform distribution of probabilities over all integers. However you could come up with many interesting non-uniform distributions. All you need to do is pick any function P that follows these rules:

• it takes integers and produces real numbers
• the reals are all between 0 and 1, and
• the sum of all the reals produced must add to 1. (Where the sum is defined as the limit of the finite sums, and so on.)

That's called the probability mass function, and with it you can answer your questions. What's the probability of picking 1? P(1). What's the probability of picking two numbers that sum to zero? That's:

P(0)*P(0) + P(1)P(-1) + P(2)P(-2) + ...

As an exercise: now think about the same question, but for real numbers. What is the equivalent of a probability mass function for real numbers? Hint: consider not the question "what is the probability of getting x?" but rather "what is the probability of getting x or smaller?" What does that function have to look like? That might help your intuition.