Choosing integers randomly
Since the question doesn’t specify a specific probability measure (which gives us the probability p_i of chosing integer i when we choose randomly), it is common to assume that p_i = p_j for all integers i, j in \mathbb{Z}, where \mathbb{Z} is the set of all integers. In other words, the probability of choosing any integer is equal to the probability of choosing any other integer.
It’s not an unreasonable line of reasoning. If we ask the same question for any finite set of integers, we can make that assumption and arrive at an answer - \frac{count-of-even-integers}{count-of-all-integers}.
Given the set of integers between -n and n, it is easy to see that the answer starts to converge to 0.5 as n starts getting large, making 0.5 the most popular answer to the question when n = \infty (set of all integers). But as it turns out, that doesn’t work.
An unequal society
Let’s assume that the line of reasoning is correct. Since the probabilities of choosing integers are all equal, let’s use the symbol \epsilon to represent this common number.
Now, if you remember the basics of probability theory, when you add up all the probabilities, they must equal to 11. However, when we add any number (except 0) to itself infinite times, the sum is always infinity no matter how small the number is. In more formal notations,
1 Technically, for triplet (X, Σ, μ) where (X, Σ) is a measurable space and μ: Σ → [0; ∞) is a measure, μ is a probability measure if μ(X) = 1.
If p_i = p_j = \epsilon for all integers i, j in \mathbb{Z}, then \sum\limits_{i=-\infty}^{\infty}p_i = \sum\limits_{i=-\infty}^{\infty}\epsilon = \infty (the sum never converges to a real number, leave alone being equal to 1).
What would you like the answer to be?
If equivalents of the fair coin or the fair dice (where all outcomes are equally likely) don’t exist for (countably) infinite outcomes2 like integers, does it mean we can never choose integers randomly?
2 When we migrate from countably infinite sets (integers) to uncountably infinite sets (real numbers), such devices are again possible (rotating wheels).
We can. Choosing randomly doesn’t mean choosing with equal probability. The probability measure can be anything we like as long as it satisfies the requirements. Let’s consider the following way of choosing integers randomly, so that, if p(x) is the probability of choosing the integer x, then,
p(x) = \begin{cases} 0 & x = 0 \\ \frac{1}{4} - \frac{1}{12} & x \in \{-1, 1\} \\ \frac{1}{8} + \frac{1}{12} & x \in \{-2, 2\} \\ \frac{1}{16} & x \in \{-3, 3\} \\ \frac{1}{32} & x \in \{-4, 4\} \\ \ldots \end{cases}
The probabilities add up to 1, and what do you know, the probability of choosing an odd integer is equal to the probability of choosing an even integer = 0.53.
3 Here is how I came up with the definition: the series \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots sums up to 1, and the alternating series (series of all odd terms and series of all even terms) sum to \frac{2}{3} and \frac{1}{3}, respectively. The definition is back-calculated from there to make probabilities of choosing odd or even integers equal.
4 Solution: x = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots = \frac{1}{3} + \frac{1}{3} * (\frac{1}{3} + \frac{1}{9} + \ldots) = \frac{1}{3} + \frac{1}{3} * x, so x = \frac{1}{2}, implying c = \frac{1}{x} = 2.
The answer can be anything we like. Here is a basic problem4 you can work through if you enjoyed it so far.
If you are curious …
All infinities are not created equal, and the :different kind of infinities we can have is itself infinite. Infinities come in different sizes (denoted by their cardinality), and intuitions break down when we travel from one cardinality to another.
The set of integers \mathbb{Z} is an example of a countably infinite set - it has infinite elements, but we can count them one by one. We can say, here is the first element, here is the second element, and so on5. We cannot do this for elements of the set of real numbers (\mathbb{R}) though, which is an example of an uncountably infinite set.
5 Cantor, the mathematician who came up with set theory, is also behind :Cantor’s diagonal argument, which demonstrate that set of integers and set of real numbers are different kind of infinities.
I digress into the nature of infinities because it gets weirder (and :way more technical) when we go from countably infinite sets (set of integers) to uncountably infinite sets (set of real numbers) - with uncountable sets we can again define probability measures corresponding to (other ways of thinking of) the size of a set!
Talking about that needs a lot of math though, so I’ll probably do it in a future post.