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If the universe goes on long enough..........

Posted By: Timothy Chow
Date: Tuesday, 18 May 2010, at 2:50 p.m.

In Response To: If the universe goes on long enough.......... (Ray Kershaw)

This is one of those annoying technicalities about probability theory. When we deal with finite sets of events, "probability zero" means the same as "impossible." However, when we pass to infinite sets, we find that it isn't quite possible to develop a formal theory that captures all of our intuitions perfectly. I won't go into the details here, but the upshot is that something has to give, and through long experience, probability theorists have found that the least painful thing to give up is the notion that "probability zero" means "impossible." Thus for example, the probability that if you throw a dart at the unit disk in the plane, it is customary to say that the probability is zero that you hit the exact center of the disk. This doesn't mean that it's impossible. After all, the dart has to hit some point, and for any given point, the a priori probability of hitting that point is zero. It's definitely weird and takes some getting used to, but in the end the theory works out better this way.

In the case of backgammon, the probability of termination is 1, even though it is possible for the game to go on forever.

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