Sunday, May 17, 2009

The Statistics of an Insomniac

Here at The Missing Piece I tend to use a statistics to clarify issues that are sometimes purposefully masked or spun by the Powers That Be to push their agenda by using their own view of the same statistics. The application of statistics to a certain problem can be both helpful and hurtful depending upon how information is presented to the reader. At the very least it can be very confusing. Game shows and other games of chance use statistics to determine outcomes all of the time. Why do I bring this up? Well, last night as I lay in bed I think I finally was able to wrap my brain around a classical statistical problem that’s been bothering me since a friend mentioned a similar situation a couple of weeks ago.

Suppose you are given a selection of three closed doors: A, B and C. Behind one of the doors is something of value to you: $1,000,000, a new car, Gillian Anderson’s phone #. Behind the other two is the booby prize: a goat, firing squad, nude pictures of your grandmother, etc. You make your initial selection based on years of test-taking knowledge and guess the good stuff is behind door C. After you make your selection one of the other doors is opened up revealing George W. Bush’s autobiography, a steaming pile of guano, or some other nonsense. Let’s say that’s door B. Then you are given the opportunity to change your selection from your initial choice (C) to the remaining door, in this case door A. Here’s the question: statistically speaking, should you switch or should you stay? Does it matter?

This is classically known as the three-doors problem. To those of you old enough to remember Let’s Make a Deal, it is also known as the Monty Hall problem, but if you’re that old then I have serious doubts that you know how to use a computer well enough to navigate to Daniel P.’s blog in this dark corner of the interweb. The statistics of the problem seem simple enough: initially each door had a 1/3 probability of hiding the raked-out chopper of your dreams. So, it makes no difference if you stay with your original choice. However, it may come as a surprise that if you are given the chance to switch selections after one is opened you should take it every time. Don’t believe me? Let’s take a look.

Assume that door A is the winner, and the other two are hiding rabid, sex-starved sliverback gorillas that have had too much to drink. There are three distinct possibilities in this scenario:

1. You choose door A. You are shown the booby prize behind one of the other doors then given the chance to switch. If you switch you get your arms pulled off, if you stay you win.
2. You choose door B, and are then shown the loser behind door C. If you switch you win, if you stay - not so much.
3. You choose door C, and are then shown your nightmare behind door B. If you switch you win, if you stay you’ll wish you hadn’t.

In only one of the above scenarios sticking with your initial selection gives you the desired outcome. In two of them switching makes you the winner. So, as I said earlier, it always makes sense to change your mind on this one. Confused? Look at it this way. Your initial selection carries a 33% chance that it is the winner. This means that the other two doors together have a 66% chance of hiding the winner. When one of those two are shown to be a loser there is still a 66% chance that the pair contains the winner, so logically this means that the remaining unopened door of the pair has a 66% percent chance of hiding early retirement. Meanwhile you’re still sitting on a measly 33% with your original selection. The same statistical reason applies for larger numbers of selections as well.

Still not convinced? Try it for yourself with a friend. Only I wouldn’t suggest the gorillas as the losers. Maybe something a little less lethal to start out with is a better idea. Then once you’ve convinced yourself that I’m right the world will be at peace again. And, yes, this is the kind of crap that keeps me awake at night. Par for the course in the Daniel P. Daniel household.

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