finite monkeys theorem

the finite monkeys theorem is a corollary to the infinite monkeys theorem which states that if you have a finite number of monkeys in a room with a finite number of typewriters, after a finite amount of time, the room will smell awful and you'll have a lot of pieces of paper that look like this:

fwo;uiheqgwuo;igawvukbvaew8yq23
f32q9y80q3g80y9q43g8iaqvpq43]
0q4goqt490]
y43q]0q4twannabananaq
439
hq3g9hqag9hwannagohomehigqaouag
qa4098gq390q32gy9qg4
y9q4ihaguoi;hgawhop
awrp
awgawrrpighaweorihgawre
iggimmeabananaq4ht9q409ygahi
aouwigefiulgwaefuligwae

in other words.. a finite number of monkeys and a finite amount of time is basically useless, as far as producing great works of literature are concerned.

To try and deal with all this crazy monkey talk I would propose a small examination into what would happen given a large but finite amount of monkeys. As much as I would like to answer the question about infinite monkeys, I can't imagine an infinite amount of monkey's, let alone figure out what to do with all that monkey shit.

So let's hypothesize for a moment. We don't want to try for the entire play of Hamlet just yet, lets just go for the phrase "To be or not to be". Count up the letters and spaces and we get 18 characters.
# of chars = 18

Perhaps it may be possible to sequester a million monkeys, that is a lot, but we are prepared to make a commitment here.
Monkeys = 10⁶

To go with our million monkeys we have a million 47 key typewriters. 10 keys are punctuation, 10 keys are numbers, 26 keys are letters, and 1 key is a space.
# of keys = 47

Let us also assume that we have specially trained monkeys, they only hit random keys and they type a startling 10 characters per second. If we have 18 characters we can give the monkeys a little extra time every now and then and assume that they type all 18 characters (or one attempt) every 2 seconds.
Single attempt = 2 seconds

The Calculation

Okay so we have basic counting principles tell us that the total number of possible outcomes for hitting one key is 47. If you're hitting two keys its 47 * 47. For 18 key presses we have 47¹⁸. Make sense so far?
# of key combinations = 47¹⁸

If you are rolling a 6 sided die, you assume that it will take 6 rolls to (almost assuredly) get any particular number you would like. So we will likewise assume that it will take the full 47¹⁸ tries to type out Shakespeare's most famous line.

Recal that it takes 2 seconds for one try at typing the line.

# of key combinations * seconds per attempt = time to complete
47¹⁸ * 2 = 2.5 * 10³⁰ seconds
Time to complete = 2.5 * 10³⁰ seconds

Okay, don't forget that there are a million monkeys trying this, so divide that time number by a million.

Time to complete / # of monkeys = time to complete for a million monkeys.
2.5 * 10³⁰ / 1⁶ = 2.5 * 10²⁴ seconds.

Almost there!

Seconds is not very useful, so lets convert to years. There are 31,557,600 seconds in a year (assuming 365.25 days in a year). So we divide again to get years.

2.5 * 10²⁴ / 31,557,600 = 7.9 * 10¹⁶ years.

The rough age of the universe, (according to the NASA WMAP project¹) is estimated to be approximately 13.7 3billion years.

7.9 * 10¹⁶ / 13.7⁹ = 4.64 * 10⁶ages of the universe.

So what does that number mean?

It means that with our million monkeys, you would have to wait

4.6 million times the age of the universe

in order to see "To be or not to be" typed out nicely on a piece of paper. Based on this estimate, I really wouldn't hold your breath for the complete works of Shakespeare.

unless we could get a billion monkeys! and maybe get rid of those 10 punctuation keys, so lets see that's 37 keys...

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¹ Nasa WMAP Mission Results - http://wmap.gsfc.nasa.gov/news/index.html