Wednesday, December 28, 2011

103: Missed It By That Much

I'm sure you've all been glued to your TV screens the past few
weeks, waiting for CNN International to announce the final results of
the Hillsboro, Oregon school board Seat 1 election.  In case you miss
their updates, here is the current count, which will most likely be
certified tomorrow:  Janeen Sollman beat me, 3582 votes to 3524, with
the distant 3rd place candidate at 2096 votes.  Yes, that's right, I
lost by a margin of only 58 votes out of over 9000 cast.  Now I know
how Al Gore felt nine years ago.
It seems like we've been hearing a lot about close elections
lately:  aside from the Hillsboro school board, some less significant
races like the Minessota senate seat and the 2000 US presidential
election have been even closer percentage-wise.  Why does this keep
happening?  One theory is that on most important issues, the majority
of the people are very evenly split, and it just happens that we're in
a society divided nearly 50-50.  However, some would subscribe to a
more cynical theory.  People are so confused by politics these days
that they just randomly vote for one of the two front-runners whose
names they recognize.  Of course, for the sake of good government, I
hope this one's wrong, but let's examine the mathematical effects of
this hypothesis for the moment.
Assuming the theory is correct, then in the close elections we are
really observing a statistical theorem in action, the Law of Large
Numbers.  This is the law that states essentially that if you sample a
random variable a large number of times, then the average number of
times you see each possible value will correspond to its actual
probability.  On in simpler terms, if you flip a fair coin enough
times, the total number of heads vs the number of tails will approach
a 50-50 ratio.  We've all probably noticed that if we flip a coin a
few times, we might get lucky and get a bunch of heads or a bunch of
tails, but in the long run the ratio will even out.  This law seems
intutitively obvious when you think about it-- in fact it was stated
without proof by Indian mathematician and astronomer Brahmagupta as
early as the 7th century AD.  But it was not until the early 18th
century that Swiss mathematician Jacob Bernoulli proved it
rigorously.  According to this law, if you flip thousands of coins,
you will expect the total numbers of heads and tails to get pretty
close.  In the show notes is a link to a website where you can
simulate large numbers of coin flips, and see results comparable to
Hillsboro's election.
But you need to be cautious when using the Law of Large Numbers--
a common misuse is what is known as the Gambler's Fallacy.  This is
the belief that somehow, the universe "knows" it is supposed to
approach the proper ratio, and therefore if you have seen a bunch of
improbable events going one way, you are likely to see a bunch of
improbable events going the other way to make up for it.  At a casino,
a gambler might see that two people rolled snake-eyes on the dice just
before him, and therefore decide he's virtually guaranteed not to roll
snake-eyes.  Or if ten of Hillsboro's random voters in a row happen to
vote for Janeen, someone deceived by this fallacy would then think
that the next ten random voters should disproportionately favor me.
But that's not how randomness works:  if people are truly deciding
their vote by a coin flip, then each voter has a 50-50 chance of
choosing either of us, just as each coin has a 50-50 chance of being
heads or tails.  The universe doesn't care about prior coin flips, or
temporary islands of improbability:  an independent event is an
independent event.  There are no divine beings standing over your
shoulder waiting to correct the coin flips in case you get too lucky.
The Law of Large Numbers is simply a statement about the natural
effects of probability, that enough random choices will even out in
the end.  So don't get tricked by the Gambler's Fallacy, and make
foolish decisions based on expecting the universe to even itself out.
And next time an election rolls around, try to really think about it
rather than flipping a coin.  Unless, that is, your thoughts lead you
to voting for an opponent of mine, in which case I won't mind a coin
flip.
And this has been your math mutation for today.

• Law of Large Numbers at Wikipedia
• Gambler's Fallacy at Wikipedia
• Web site for simulating large number of coin flips at Utah State