When a concept of physics or mathematics is too difficult for the regular
layman to understand, someone will come up with an *analogy* close to
real life to describe this concept or phenomenon. If this analogy is catchy
enough, it will then basically be engraved in stone, so to speak, and people
will start repeating it blindly, without really understanding it.

The problem with analogies is that they are usually very weak representations of the physical or mathematical phenomena behind them. This means that they can usually be understood in the wrong way and all kinds of misconceptions may arise. Often the wording of the analogy is not precise enough and people get the wrong ideas.

Sometimes the analogies themselves are simply bad and do not correspond
to their underlaying theory very well, if at all. Sometimes analogies are
simply *wrong*, period.

One widely spread analogy which, in my opinion, at least borders the wrong is the one related to monkeys, typewriters and Shakespeare. There are actually slight variations of this analogy, changing the number of monkeys and the amount of time, but one typical version goes like this:

A million monkeys hammering a million typewriters for a million years will eventually write the entire works of Shakespeare.

Some versions try to fix the fallacy in that statement by changing "a million years" with "an infinite amount of time".

What this analogy is actually trying to say is that given an evenly
distributed (truly) random number generator, popping an unlimited amount
of values from it will make the probability of the appearance of any given
finite subsequence unlimitedly high. (Note that this is not the same as
"any given finite subsequence *will* eventually appear.")

However, the analogy is horribly flawed. There are numerous errors in it:

Using precisely *monkeys* in the analogy is an extremely poor choice.
This is because "monkeys hammering typewriters" is a very poor random number
generator.

The theory needs an *evenly distributed* random number generator.
This means that all values have the same probability. An animal hitting a
typewriter is probably one of the poorest choices for this. It may well
be that some letters (like 'A') are never or very rarely hit while others
are being hit almost constantly.

Even if we assumed that the monkeys were a true evenly distributed
random number generator, why precisely a million years is enough for the
works of Shakespeare to pop up? There's certainly no law of mathematics
or physics that says this. It's perfectly possible for the monkeys to
hammer their typewriters for a million years and *not* come up
with the works of Shakespeare, even if they create truly random sequences.

This is the reason why some people try to fix this problem by changing it to "an infinite amount of time". However, that doesn't leave the analogy flawless either:

If we assume that a monkey is a true evenly-distributed random number generator and that it had an infinite amount of time, why do we need a million of them? One would be enough. It doesn't make any difference.

However, even that is not completely flawless:

Even if we had a true evenly-distributed random number generator and
an infinite amount of time, stating that it *will* eventually
generate the entire works of Shakespeare is wrong. It *might*
generate them, but there's no guarantee.

Why is it wrong? It is wrong for the simple reason that there exists an
infinite amount of random sequences *not* containing the entire works
of Shakespeare. It is possible for the random number generator to go through
those sequences before it generates Shakespeare's works. Since there's an
infinite amount of other sequences it will thus take an infinite amount of
time to go through them and the works will thus never be generated.

Thus, it is *possible* for the works of Shakespeare to be
generated, and the probability is actually unlimitedly high, but there's
no absolute *guarantee* that it will happen. It is thus wrong to
say that "it *will* eventually generate the works of Shakespeare".

The misconception that the works of Shakespeare *must* appear
at some point is actually closely related to the so-called
*gambler's fallacy*. This fallacy is the mistaken conception that
past events affect future probabilities. The classical example is coin
tossing: If we are tossing a coin repeatedly and it has given us 9 heads
already, the fallacy is to think that the probability of getting tails
in the next toss is very large. Naturally this is not so: The probability
of getting tails in the next toss is still 50%. Past tosses do not affect
this probability in any way.

The monkeys-typing-Shakespeare statement can be simplified to a coin-tossing statement: "If we toss a coin an infinite number of times, then getting tails must happen at some point."

This is a fallacious statement. Sure, the probability of getting an
infinite number of consecutive heads is infinitely small, but each toss
still has the 50% chance. There's no law of mathematics which would say
that in repeated tosses a certain solution *must* appear. Yes,
it's *extremely likely* that it will appear, but there's no law
that says that it definitely *must* appear. It's perfectly
*possible* that we get an infinite number of heads. Each toss
has 50% of probability, and past tosses do not affect this.

In order to avoid some confusion, let me clarify a few things:

What the monkey analogy is trying to describe is, basically, the property
of the mathematical definition of an evenly-distributed random number
generator. This definition is closely related to the concept of
*infinity*.

Basically, what it's saying is that if we have an *infinite*
sequence of evenly-distributed random characters (including spaces to
separate words, etc), the entire works of Shakespeare must appear in
that sequence (an infinite number of times, no less). If they didn't appear,
then the randomness would not be evenly distributed, by definition. If the
works didn't appear, then there would be some kind of bias in the random
number generator, making it not evenly-distributed.

Where the analogy fails is trying to bring the concept (related to infinity) into a concrete physical form. If you take any finite portion of that sequence, no matter how large, the probability of Shakespeare's works appearing in that sequence is less than 1.

If you had an evenly-distributed random number generator producing
random characters, it may never produce Shakespeare's works, no matter
how long you wait. You can expand the sequence as much as you want,
without any upper limit, and there would still be no guarantee that the
works will appear. As stated, each new character produced by the generator
is *not* affected by previously produced characters, and the probability
for each new character to break the desired outcome is always the same.
It doesn't matter how many times you take a new random character, the
probability doesn't change.

The concept of *infinity* that the mathematical definition is
describing is a more abstract concept than can be described with any
real-life analogy. The analogy wrongly presents the notion that a physical
random number generator must eventually produce the entire works of
Shakespeare. This is not so: No matter how long you run the generator,
there is no guarantee.

It is *possible* for the generator to produce the works. And in
fact, the probability of this increases with the amount of generated characters.
In other words, if you were to set the generator to generate a trillion
random characters, the probability would be larger than if you set it to
generate a half trillion. However, no matter how long you run it, the
probability will never reach 1, and there will never be any guarantee.
You could literally wait *forever* and never get the works.