"Angels can fly because they take themselves lightly." – G. K. Chesterton

Archive for the ‘Mathematics’ Category

Monkeys and Typewriters

for some reason there’s a lot of confusion about the topic of monkeys and typewriters.  I’m not sure how it originated.  Sometimes in the late 19th or early 20th century, people started believing that if you put some number of monkeys at typewriters, they’d eventually produce something.

A monkey

 

 

I’ve heard variation in both the numbers and the text to be produced.  Some versions say 13 monkeys, others say 700 or 5,000.  Some say they’ll produce the complete works of Shakespeare, while others say it’s the U. S. Constitution or Moby Dick.  None of them are very precise on the details.

A typewriter

In fact, there are many problems with this statement.  First of all, a monkey at a typewriter would not be guaranteed to type anything.  It wouldn’t even be guaranteed to stay at the typewriter.  If it was guaranteed to type, it still might not type all of the keys, but might instead limit itself to a small subset of them.  And worst of all, it would eventually die.  Monkeys tend to do that.

A chimpanzee (which is not a monkey) at a typewriter

Suppose we lay aside those difficulties for a moment.  Imagine we had a monkey that would type diligently at sixty keystrokes per second, every second.  Since there are 26 letters in the alphabet–we ignore punctuation and capitalization–this means that there’s a 1in-26 chance that the first letter would be ‘c’.  Likewise there’s a 1-in-26 chance that the second letter would be ‘a’.  Since 26 multiplied by 26 is 676, there’s a 1-in-676 chance of the monkey producing ‘ca’ together.  Extending the process a little bit, we find a 1-in-456,976 chance of it typing the word “call”.  And there’s a 1-in-64,509,974,703,297,150,976 chance of it writing the sentence “Call me Ishmael.”

The Complete Works of Shakespeare

Alternately we could think of it like this.  Even our very diligent hypothetical monkey would have to type for more than a trillion years before the odds of it typing “Call me Ishmael” were great than one half.  Given that time frame, I suggest we dump typewriter-equipped monkeys once and for all.

 

And a very happy Pi Day to you

Today is Pi Day.  In case you’re not familiar with this newly minted holiday, it’s based on today’s date in numerical form: 3/14.  Now the number Pi, which is the good old ratio of a circle’s circumference to its diameter, begins with the digits 3.14 and then continues.  Here’s a link to a billion digits of pi if you want them.

 

I could proudly say that I got into Pi Day on the ground floor, so to speak.  Way back in March of the year 2000, I was a senior in high school, and a very geeky senior in a very geeky high school at that.  My friends and I were just beginning to explore the recesses of the internet, using Netscape and AltaVista and other things that today’s youngsters have probably never heard of.  One of us stumbled across a website devoted to the number Pi, featuring sonnets and other poems to Pi, the Hunt for Intelligent Life in Pi, and advocacy for the holiday of Pi Day.

Ah, we were young and innocent in those days.  We thought that the idea of spending a day celebrating a transcendental number was simply the greatest thing, a perfect way to show off our geekiness and eccentricity and the dedication to math and inside jokes that defined our identity as separate from the hoi polloi.  Little did we guess that in a few years, Pi Day would go mainstream.  That every school in the country would soon be hosting a celebrating of this day, complete with demonstrations involving hula hoops and consumption of many flavors of pie.

Yes indeed, our private little geeky activity has no been swallowed up by America’s mainstream culture.  It no longer outlines the brainy set as distinct from everyone else.  If today’s high school math nerds want a holiday to show off their differences, they’re going to have to pick a different non-repeating decimal.  I would suggest February 24.  (It’s square root of 5 day, needless to say.)

Bracketology

Here is on of my few posts on sports.  Right now is “March Madness”, and millions of people all over the country are scrambling to fill in their brackets and predict the winners of the 63 games that make up the NCAA Tournament.  (Actually it’s 67 games, but never mind that.)  Various strategies are on the table.  There’s the strategy of picking winners by their mascots: large animals beat small animals, humans beat animals, armed humans beat unarmed, and so forth.  Then there’s the strategy of choosing teams based on their time zone: teams from the Eastern Time Zone will have more time to rest up when they travel west, while those traveling west to east will be short on sleep.  I, on the other hand, have a strategy that actually works.

The strategy is this: pick the higher-seeded team in every game.  In other words, don’t predict any upsets.  Thus the only real choices you have to make are in the Final Four, since you’ll predict all the #1 seeds to reach it.

“Wait!”  I hear someone crying.  “That doesn’t make sense!  There are always upsets, so if you want to win the pool, you have to predict upsets.”  Not so.  That reasoning is based on a misunderstanding of probability.

Consider a piece of advice I read on a website last year.  On average, one 11th seed defeats a 6th seed in the opening round each year.  (There are four games pitting 11th seeds against 6th seeds.)  Thus, this hapless website advises readers to pick one upset in which and 11th seed defeats a 6th seed.

That advice is flawed, though, because you don’t know ahead of time which 11th seed will pull off the upset.  The odds are against a victory by any one particular 11th seed,  even if they’re in favor of one of the four pulling off the upset.  It’s an exercise in basic probability theory to show that your expected returns are better if you wager on all the 6th seeds emerging victorious.

The same logic applies to the tournament at large.  While there will certainly be some upsets, you can’t know ahead of time which possible upsets will occur, so you don’t benefit from predicting upsets.  Predicting the higher seed every time and the odds will be in your favor.  That’s my advice, in any case.  If you choose to pick by mascot, don’t blame me for the results.

Tag Cloud