Wizards of Oz

"Life is fraughtless ... when you're thoughtless."


Large Numbers

A famous thought experiment postulates that a monkey, strumming unintelligently on a typewriter for an infinite amount of time, would eventually create all of the works of Shakespeare. Although often attributed to T.H. Huxley, a 19th century English biologist, it is a metaphor used in a 1913 essay by Émile Borel to describe large, random sequences of numbers.

During the 2007 Boyd Conference in Quantico, Virginia, LtGen(ret) P.K. Van Riper, USMC, described the rapidly-escalating variance in chess moves. Even though the chess board is a tightly constrained "battlespace" (8x8 grid, 32 pieces), after five moves there are more than 800,000 possible combinations. After six moves, this number increases to more than 9,000,000. And Wolfram's MathWorld shows that the possible number of positions after 40 moves is more than 1E120 -- a "1" followed by 120 zeros. This is nearly a billion-trillion times larger than the number googol (1E100, or the inspiration for the unintentionally-misspelled Internet search engine Google).

To give this number context, scientists today postulate that there are only 1E80 particles in the visible universe. And the age of the universe is estimated at 1.4E10 (14 billion) years.

So let's go back to our monkey. As an undergraduate physics major at Berkeley, one of the first homework problems in my thermodynamics class was a variation of the "infinite monkey theorem": we had to determine the probability of a trillion monkeys, typing randomly without pause at 10 keys per second, to randomly type the words of Hamlet. By assuming Hamlet was comprised of approximately 100,000 characters, and that a typical keyboard has 40 keys (without regard for punctuation or capitalization), the probability of a random string is 1/40 * 1/40 * 1/40 ..., repeated 100,000 times.

Despite having a trillion (i.e., 1E12) monkeys typing continuously at 10 keys per second, our solution was that it would still take more than 1E1000 years -- in other words, nearly googol (1E100) times the age of our known universe -- before reaching a 50% probability.

This is important for anyone charged with analysis or decision making responsibilities. We live in a world where just three significant figures (e.g., 99.9%) is considered accurate enough, and "six-sigma" (six significant figures, "1-in-a-million") is the ultimate achievement in performance. Too often we overlook the dynamics of our complex world, and we tend to dramatically underestimate variance in subsequent effects of actions.

So, if someone suggests to you that they can predict future actions in, say, a battlefield, just remember these facts:
  • The number of chess moves after a 40-move game is 1E120
  • The fastest computers in the world process about 1E15 operations per second
  • There are 1E80 particles in the visible universe
  • We still can't predict the weather accurately -- and nature isn't trying to deceive us!
Caveat emptor...

(H/T to Zenpundit for the post idea.)

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At 25/7/07 07:25 , Blogger Sean Meade said...

postulate: the physical world in complex. as we move into psychology, including choice, complexity increases rapidly*.

*tempted to write 'exponentially', but that would be imprecise, and i don't want to be accused of innumerancy by Mark! ;-)

At 2/10/09 15:58 , Blogger Sbiis Saibian said...

In your article you mention that the solution to your homework problem was over 10^1000 years.

You state that this is "nearly" 10^100 times the current age of the universe. However that would only amount to about 10^110 years.

When you multiply powers of ten together you add the powers, not multiply.

so 10^100 x 10^10 does not equal 10^1000, but "only" 10^110.

This does illustrate just how deceptive large numbers really are.

10^1000 is roughly the current age of the universe (10^10 yrs) raised to the 100th power ! Just imagine what that means. The entire history of the universe would be dwarfed by a factor of itself 99 times !!

Things get even more counter-intuitive when dealing with double exponential numbers such as the Googolplex (10^10^100).

-- Sbiis Saibian

At 3/10/09 09:18 , Blogger deichmans said...

Sbiis, Yep - careless mistake on my part. Thanks for the correction! -s


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