Tuesday, December 25, 2012

Almost Surely

The concept of probability when dealing with infinite spaces can be a little confusing and counter-intuitive. One "paradox" that came up while I was watching lectures for a computational finance course on Coursera was the idea that an event which has probability zero of occurring can actually occur. For example, if you choose a random real number between 0 and 1, the event of choosing 0.5 exactly can certainly occur, but any reasonable calculation of the probability will lead to a value of zero. Naturally, I assumed people had formalized this notion, so I did a bit of searching online.

This led me to the concept of events that happen almost surely, which is to say they occur with probability one; alternatively, events that happen almost never have zero probability of occurring. There's nothing too crazy here; an event that happens almost surely simply has an infinitesimally small chance of not happening. So for mathematical purposes it only makes sense to assign the probability as one. To quote the Wikipedia article, "the difference between an event being almost sure and sure is the the same as the subtle difference between something happening with probability one and happening always."

Examples of events that happen almost surely are: flipping a coin an infinite number of times and getting tails at least once, picking a random real number between 0 and 1 and getting an irrational number. Examples of events that happen that happen almost never are: a random variable drawn from a continuous probability distribution taking a particular value, picking a random real number between 0 and 1 and getting 0.5. Moreover, the complement of an almost sure event happens almost never and vice versa. So, to sum things up, an event that happens with probability zero might not never happen, just almost never happen.

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