The monkey on a typewriter
Alfred is a monkey who lives in the local zoo. He has a battered old typewriter
with 26 keys for the letters of the alphabet, a key for a full stop, one for a
comma, one for a question mark and one for a space – 30 keys in all. He sits in a
corner filled with literary ambition, but his method of writing is curious – he hits
the keys at random.
Any sequence of letters typed will have a nonzero chance of occurring, so
there is a chance he will type out the plays of Shakespeare word perfect. More
than this, there is a chance (albeit smaller) he will follow this with a translation
into French, and then Spanish, and then German. For good measure we could
allow for the possibility of him continuing on with the poems of William
Wordsworth. The chance of all this is minute, but it is certainly not zero. This is
the key point. Let’s see how long he will take to type the soliloquy in Hamlet,
starting off with the opening ‘To be or’. We imagine 8 boxes which will hold the
8 letters including the spaces.
The number of possibilities for the first position is 30, for the second is 30,
and so on. So the number of ways of filling out the 8 boxes is 30 × 30 × 30 ×
30 × 30 × 30 × 30 × 30. The chance of Alfred getting as far as ‘To be or’ is 1
chance in 6.561 × 10^11. If Alfred hits the typewriter once every second there is
an expectation he will have typed ‘To be or’ in about 20,000 years, and proved
himself a particularly long-lived primate. So don’t hold your breath waiting for
the whole of Shakespeare. Alfred will produce nonsense like ‘xo,h?yt?’ for a great
deal of the time.
How has the theory developed?
When probability theory is applied the results can be controversial, but at least
the mathematical underpinnings are reasonably secure. In 1933, Andrey
Nikolaevich Kolmogorov was instrumental in defining probability on an axiomatic
basis – much like the way the principles of geometry were defined two millennia
before.
Probability is defined by the following axioms: