Counting infinities

Choosing integers randomly

If you choose an integer randomly, what is the probability that the chosen integer is even?

Since the question doesn’t specify a probability measure (which gives us the probability $p_i$ of choosing 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:

Given the set of integers between $-n$ and $n,$ it is easy to see that the answer starts to converge to $0.5$ as $n$ gets large, making $0.5$ the most popular answer to the question when $n = \infty$ (the 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 $1.$Technically, for a triplet $(X, \Sigma, \mu)$ where $(X, \Sigma)$ is a measurable space and $\mu : \Sigma \to [0, \infty]$ is a measure, $\mu$ is a probability measure if $\mu(X) = 1.$ However, when we add any number (except $0)$ to itself infinitely many times, the sum is always infinity, no matter how small the number is. In more formal notation:

If $p_i = p_j = \epsilon$ for all integers $i, j$ in $\mathbb{Z},$ then

$$\sum_{i=-\infty}^{\infty} p_i = \sum_{i=-\infty}^{\infty} \epsilon = \infty$$

The sum never converges to a real number, let alone equals $1.$

What would you like the answer to be?

If equivalents of the fair coin or the fair die (where all outcomes are equally likely) don’t exist for (countably) infinite outcomesWhen we migrate from countably infinite sets (integers) to uncountably infinite sets (real numbers), such devices are possible again (rotating wheels). like integers, does it mean we can never choose integers randomly?

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

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: $0.5.$Here is how I came up with the definition: the series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$ sums up to $1,$ and its alternate terms (the series of all odd terms and the series of all even terms) sum to $\frac{2}{3}$ and $\frac{1}{3},$ respectively. The definition is back-calculated from there to make the probabilities of choosing odd or even integers equal.

The answer can be anything we like. Here is a basic problemSolution: $x = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots = \frac{1}{3} + \frac{1}{3}\left(\frac{1}{3} + \frac{1}{9} + \cdots\right) = \frac{1}{3} + \frac{1}{3}x,$ so $x = \frac{1}{2},$ implying $c = \frac{1}{x} = 2.$ you can work through if you enjoyed it so far.

If you are curious…

All infinities are not created equal, and the number of different kinds of infinitiesAleph numbers measure the sizes of infinite sets. The smallest, $\aleph_0$ (aleph-null), is the size of the integers. The real numbers form a strictly bigger infinity. And there is no biggest one: Cantor showed that the set of all subsets of any set is always larger than the set itself, so the sizes of infinity go on forever.Aleph number on Wikipedia 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 infinitely many elements, but we can count them one by one. We can say, here is the first element, here is the second element, and so on.Cantor, the mathematician who came up with set theory, is also behind Cantor’s diagonal argumentSuppose someone hands you a list that claims to contain every real number between $0$ and $1,$ written as infinite decimals. Build a new number by going down the list and changing the $n$th digit of the $n$th number. The new number differs from every number on the list in at least one digit, so it isn’t on the list. No list can hold them all: the real numbers can’t be counted one by one.Cantor’s diagonal argument on Wikipedia, which demonstrates that the set of integers and the set of real numbers are different kinds of infinities. We cannot do this for the elements of the set of real numbers $(\mathbb{R})$ though, which is an example of an uncountably infinite set.

I digress into the nature of infinities because it gets weirder (and way more technicalA probability space is a triple $(\Omega, \mathcal{F}, P):$ the set of possible outcomes $\Omega,$ a collection $\mathcal{F}$ of events (subsets of $\Omega)$ we are allowed to ask about, and a function $P$ that gives each event a probability, with $P(\Omega) = 1.$ On uncountable sets like the real numbers, some subsets can’t be given a sensible probability at all, which is why the collection of events has to be chosen with care.Probability space on Wikipedia) when we go from countably infinite sets (the set of integers) to uncountably infinite sets (the 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.