How can any statistics be calculated from a infinite population.

Question

My understanding of some statistics:

For a given experiment a finite number of samples can be taken, defining the sample size. However this experiment may have an infinity large population size.

Given that in an infinitely large population size any possible outcome will occur an infinite number of times, how can we truly say one outcome is more likely than another. Mathematically lets say from our sample we've found that there is a 0.1 chance of drawing a blue ball, and a 0.0000001 chance of drawing a red ball. If the balls are always replaced these odds do no change. However since the ball can be drawn any number of times this will happen:

blue ball count: 0.1 * x red ball count: 0.0000001 *x

where x is any number. Clearly the odds do not change with increasing x. But if x is infinitely large then:

blue ball count: 0.1 * infinity = infinity red ball count: 0.0000001 8 infinity = infinity

So how can we confidently state any odds when the population size is infinity?

P.S. This is just out of interest so I would appreciate as much detail, links etc. as possible

Answer 1

Practically speaking, there cannot be an infinite population of balls - there isn't room in the universe.

However, your multiplying infinity by .1 doesn't relate to the problem.

A sample is random if any combination of objects is equally likely. This is possible even with an infinite population. For example, if the balls are numbered $1 - \aleph_0$ then what is required is that, in a sample of size 3 (for example) the probability of picking

1, 2, 3

is the same as

100020101, 2912091, 209101100021

Answer 2

I think your confusion stems from seeing probability as based on a number as opposed to a relative frequency. If you actually drew an infinite sample with replacement, then determining the probability via counting is pointless, as you have shown.

However, you can use the frequentist notion of probabilty (see 1, 2) to get an answer:

Let R = Draw a Red Ball, B=Draw a blue ball, then the relative frequency definition of probability is:

$P(R) \equiv \lim\limits_{n_{R+B}\rightarrow \infty} \frac{n_{R}}{n_{R+B}}$

What this definition says is that its the relative fequency of the occurance of red vs. blue draws that defines the probability. Your infinite sample size experiment merely takes the relative frequency notion of probability to its logical extreme, and will result in the calculation of the exact relative frequency of blue balls to red balls, even though both are infinite.

The actual numbers you are using are confusing, because they don't add to 1. However, if you state that there is a 0.1 chance of getting a red ball, then you would expect to get 1 red ball out of every 10, or, in the limit, your proportion of red balls will approach 0.1.