infinite monkey problem - probability of an infinite sequence containing an infinite sequence [duplicate]

Question

Note: This question is specifically about when the infinite monkey theorem is extended to reproducing an infinite sequence (as oppose to a finite one)

I was browsing wikipedia, and came across the infinite monkey problem. I understand that if Random keys are selected for an infinite period of time, eventually any finite combination of characters will eventually be produced. For example, a complete copy of Hamlet. However, I am confused about what happens when this is extended to infinity. According to wikipedia:

If the monkey's allotted length of text is infinite, the chance of typing only the digits of pi is $0$, which is just as possible as typing nothing but Gs (also probability $0$).

http://en.wikipedia.org/wiki/Infinite_monkey_theorem#Almost_surely

However, this suggests that the probability of typing any combination of characters (that are infinitely long) is also $0$. However, the probability that a combination is typed is 1. However the sum of all possible events appears to be $0$, not $1$ (as the probability of any combination of infinite length being typed such as pi is $0$ as $0+0+0+0+...+0=0$

I'm not sure if I have just misunderstood this problem or if I have made a false assumption. It might have something to do with concept of "Almost certainly", but I was a bit intimidated by the wikipedia article.

Answer 1

Here is a similar example: let $x$ be a number chosen uniformly from $[0,1]$. For every $y \in [0,1]$, the probability that $x = y$ is $0$; and yet $x$ must be equal to something! What goes wrong here is that there are uncountably many numbers in $[0,1]$, while the uniform measure over $[0,1]$ is only countably additive. Measures with stronger additivity properties are studied in measure theory (in the set-theoretic parts), but the usual measures one studies in probability theory are only countably additive (or, as it is more commonly known, $\sigma$-additive).

Answer 2

The probability of typing out, for example, the digits of pi, is $0$. But, there are an uncountably infinite number of strings of characters the monkey could type out, so the total probability of all events is still equal to $1$. It's a bit like saying if you could throw a dart at the real number line, there is no chance you could hit one specific number. The real numbers are uncountably infinite.

Answer 3

It's just one of those complications that comes up when you deal with infinity.

Here's how I think about it. When you want the monkeys to type out Hamlet, you can figure that Hamlet only has a finite length and you can compute the probability of randomly typing it. Since the monkeys are typing out an infinite string of text, you can divide that up into infinitely many Hamlet-length chunks and reason that one of those is bound to be Hamlet.

The trouble with asking them to type out an infinite string is that this no longer works. Once you take out an infinite piece from the monkeys' output, you only have finitely many characters left. So the argument that worked for Hamlet doesn't work for an infinite string.

The odds of any particular infinite string being chosen is zero. What this means is that if you repeated the experiment (countably) infinitely many times, you still wouldn't expect that string to come up, since the number of possible strings is of a higher order of infinity. This also means that adding up all the probabilities no longer works the same way when you have uncountably many events to choose from.