One way to understand algorithmic probability is by using a famous metaphor involving a
monkey and a typewriter.
The infinite monkey theorem introduced by Emile Borel in the context of randomness,
is a metaphor for the probability of random events to be capable of producing structured
sequences such as words.
The theorem states that a monkey hitting keys at random on a typewriter can produce any
text no matter how long and sophisticated such as the complete works of William Shakespeare
given enough time.
But if instead of a typewriter one places a monkey in front of a computer with each
key on the keyboard representing an instruction in binary for a computer program, it turns
out that the probability of the monkey to produce the complete works of Shakespeare,
or any object of high complexity, increases dramatically at the point that it might be
likely to happen during the span of time of the age of the universe!
This piece of code illustrates the idea.
Indeed, according to algorithmic probability, outputs encoding information such as Shakespeare's
plays that are far removed from random text have greater chances to be the output of a
random computer program because its producing program would be shorter than something producing
pure randomness.
In other words, the less random---and therefore more compact to be described---the higher
the probability.
What algorithmic probability tells is that it is way easier to try to reproduce something
such as life and humans to produce something like Hamlet in, as we know, much less than
the age of the universe, as the result of a sophisticated physical, chemical and biological
process than trying to reproduce Hamlet from scratch letter by letter that takes much longer
than the age of the universe.
This small computer program illustrates how a short random computer program is capable
of producing highly structured non-random outputs with much greater chances than what
classical probability theory would establish.
The exact algorithmic probability cannot really be computed but approximated.
A lower bound using Shannon entropy tells that the probability of the programmer monkey
to hit the target binary sequence is 1/(the shortest program producing the string) which
cannot be shorter than the logarithm in base 2 of the string length and should be quite
close if the string is highly compressible or not random.
This computer program illustrates that this is actually the case, and that the number
of keystrokes is rather accurate.