March 31, 2014

Mapping the Possibility Space

Near the end of one of my favorite movies, Magnolia, it abruptly starts raining frogs.   Cars crash, people freak out; it's a full-blown crisis. 

One of the characters, a young boy, is looking out the window, taking it all in.  Then he says something both simple and utterly profound:

"This is something that happens.  This is one of the things that happens." 

Statisticians have a name for this category -- this category of "things that happen" ...or, at least "things that could happen".  It's called the Possibility Space.  If something can happen, no matter how unlikely it is to happen, then it resides within the Possibility Space.

Unlikely Is Not Impossible
This concept conflicts with how people typically think.  Normally, if an event is sufficiently unlikely, people will consider it impossible.  Indeed, it doesn't even have to be particularly unlikely for most people to write it off.  

A lot of people will say things like "it's impossible for the Bengals to win the Super Bowl this year."  Oh puhleeze.  It's not.   It may be impossible for the Red Sox to win the Super Bowl this year (...being that they're a baseball team and all), but even that is probably not impossible -- probably more like a one-in-a-duodecillion chance.

Pondering The Edge
I spend a lot of time wondering what fantastic things might exist at the very edge of the Possibility Space -- events that are exceedingly unlikely, but still possible.

For example, consider your personal Probability Space.  Does it include you...

    ...wining a Nobel prize?
    ...playing free safety in a 2019 Jaguars Super Bowl win?
    ...becoming president of the United States?
    ...being nominated as the next Pope? for a million years?
    ...traveling to Mars?
    ...witnessing a true divine-intervention miracle?
    ...transmuting lead into gold?
    ...speaking with an extra-terrestrial intelligence?

Granted, none of these things is likely.  But are they impossible?  No, none of them are probably impossible.  They are in your Possibility Space, even if they're just a glimmer in an ocean of outcomes.

Beyond The Edge
Tesseract: a 4th-dimension cube
Conversely, what things are forever out of reach, no matter how the cards fall? What things are beyond the Possibility Space
    ...traveling faster than the speed of light?
    ...understanding how the universe works?
    ...visualizing the fourth dimension?
    ...observing the surface of a black hole?
    ...traveling back in time?
    ...communicating with spirits?
You are incredibly unlikely.  Yet, you exist.
You might be wondering what's the point in thinking about unlikely-but-not-impossible things.  I mean, they're not going to happen, right?

No, not right.  As it turns out, if you look things from the proper perspective, virtually everything that has ever happened has been ferociously, mind-bendingly unlikely.  Rewind to 10,000 B.C. -- what is the likelihood that your family tree would evolve in the way that it did?  Well...if we assume that each generation is twenty years (it's probably smaller than that, but whatever), then that's six hundred generations ago.  What's the likelihood that your ancestors would (a) survive long enough to mate, (b) meet just the right partner, (c) mate at just the right time, and (d) have just the right child... 

I tried to calculate it, but...I'm not able to express it in words.  The biggest named number is a British centillion (10^600), and I blew through that pretty quickly.

The Most Unlikely Outcome...
My brother is often going on about coincidences.  He'll spot a common pattern between his license plate, the first few digits of some girl's phone number, and the age of his cat, and before I know it, he's calculating how vastly improbable this coincidence is.  I'm rarely moved.

Why?  Because, while it's true that any one set of circumstances is unlikely, there are so many unlikely things that could happen, that some are highly likely to occur.  For a moment, ponder the virtually infinite number of coincidences that could occur:
  • You and your plumber have the same first name.
  • Your dog was born on the same day as your neighbor's son.
  • The last three streets you lived on all started with the same letter. 
You could spend a lifetime thinking of coincidences, and not scratch the surface.   So, all told, the most unlikely outcome in the that only likely events will occur.

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