Friday, August 12, 2011

The landscape of probability

We all know that the probability of some condition can lie anywhere between a sure thing (which we represent as a probability of 1) and a flat-out impossibility (0). But it turns out there are several other points of interest along the way. Let's take a tour.

When we say that something is a sure thing, we mean it is bound to be so. For example, the probability that a bachelor is unmarried is 1. This is a logical sure thing because, by the definition of 'bachelor', it couldn't be otherwise. (It could also be called an apodictic sure thing, however that's pretty much guaranteed to sound pretentious—but I'm getting ahead of myself.) Now consider the statement that an object with a positive electrical charge is attracted to an object with a negative electrical charge. This is a physical sure thing: though there may be a universe where this isn't true, it is true in ours.

Now let's move from sure things to things that are pretty much guaranteed. For example, it's pretty much guaranteed that the sun will rise tomorrow. Not a sure thing though: who knows what strange astronomical events might come to pass overnight? More about this in moment.

Nevertheless, most of the time, when we think about things that we consider likely, we're not thinking of things that are so overwhelmingly likely as the sun rising tomorrow. Likely things just have better than even odds.

Even odds (a probability of ½) is of course the sweet spot where the probability of some condition is exactly equal to the probability of its opposite. This can be interpreted as an expression of perfect uncertainty about the condition, and this was how Pierre-Simon Laplace used it in working out a solution to the Sunrise problem.

But as we continue our stroll, we find ourselves in the realm of unlikely things. Note that they don't have to be terribly unlikely. Something with a probability of 0.49 is (just slightly) unlikely, in that it has worse than even odds.

As the probabilities get thinner and thinner, we soon find ourselves encountering things that ain't gonna happen. These are the opposite of sure things. I'd love to win $50,000,000 in the lottery, but it ain't gonna happen. Well, of course it is gonna happen, but not to me (in all likelihood). Not that it's a physical impossiblity of course, much less a logical impossibility.

I'd just have to be awfully lucky.


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Blogger Oscar said...

I came from Rationally Speaking.

Nice post, man. And nice picture.

6:23 PM, August 20, 2011  
Blogger Nick Barrowman said...

Thanks, Oscar!

9:30 PM, August 20, 2011  
Blogger PB said...

A point you made that I'd never considered before - 50% as being "perfectly uncertain".

Most laypeople would think that something being "even odds" happening would not be better odds than "perfectly uncertain" - but they're really the same thing!

Funny how language can lead us astray.

9:50 AM, August 22, 2011  
Blogger Nick Barrowman said...

Absolutely. Another way to think about a 50% probability is as the point of maximum entropy, that is the situation in which the actual outcome is the most informative.

11:09 PM, August 22, 2011  
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