Friday, April 15, 2011

Probably Not

Bear with me, because we are going to start out with a little bit of easy mathematics. I promise that there is a more interesting point. Anyway, one way to think of the probability of something is how often you expect it to happen. So, if something happens 50% of the time, like a fair die rolling an even number, out of one hundred rolls you would expect about 50 of them to be even. Similarly, if you have a .5% chance of winning a raffle, you expect that you would win about once if the raffle happened 200 times, or about 5 times if it were to happen 1000 times.

Something with a 100% probability is, for practical purposes, a sure thing, and something with 0% probability is, for practical purposes, impossible. Because mathematicians are lazy, something I spent quite some time emphasizing to my class, we often call things that happen with 0% probability, 0 probability events. Since we assume that something does happen, the probabilities of all the possible different outcomes needs to add together to 100%.

As mentioned, 0 probability events are impossible for practical purposes, but with a bit of consideration one can come up with times when they would happen. Suppose you have a computer that picks a positive whole number at random out of all the positive whole numbers, with each number being equally likely. There are so many positive whole numbers that the chance of any individual number being picked is 0%, yet as soon as the computer picked one, a 0 probability event would have happened. Of course, this is impossible to implement in reality, because no computer has enough memory or processing power to choose a whole number from amongst ALL of them, but it is an example of a 0 probability event occurring, theoretically.

A similar 0 probability event is for a number, not necessarily a whole number, between 0 and 1 to be picked assuming each number is equally likely. Again, because there are so many numbers between 0 and 1, each individual number has 0 probability, but also again, this also means no computer could actually pick amongst them. So, the question becomes, can 0 probability events occur non-theoretically?

Asking this question prompts some interesting observations about our world. Suppose you think of a dart board as a continuous surface, then throwing a dart at it and seeing where it sticks is a 0 probability event, as there are just so many slightly different places it could have stuck. However, if you think of the dart board as being discrete (made up of discernible blocks), even if you have to go to the atomic level, then each separate place the dart could land has some non-zero probability. Admittedly small, but more likely than impossible. So, refusing to believe that impossible events happen forces one to consider a discrete universe, which is hardly a handicap because at its basic level the universe does seem, at this point, to be discrete.

Also interestingly, consider the concept of free will. If everything in the universe is deterministic, that is what happens next is forced to happen because of what has happened previously, then what happens is the only thing that could happen and has 100% probability. Of course, this model of the universe is not conducive to a belief in free will. If multiple things are possible, but only a finitely many number at any given juncture, then free will is theoretically possible, allowing us to choose between the provided options, and each possible option will have positive probability. However, if anything is possible, as they sometimes say, then it becomes mathematically viable for possible events to have zero probability. It is not necessary, because you can add up an infinite number of positive numbers and still only get 100%, but under these conditions it becomes possible.

So, if one wants a 0 probability event to occur, one must believe that anything is possible. I find that a bit interesting. Also interesting, this thought process occurred because, on my way home from school, I came up with a joking characterization of Occam's Razor for the Middle Ages, "the explanation that requires the fewest witches must be true." Then, to be a bit more serious, I reformulated as, "the explanation that requires the fewest 0 probability events must be true." That got me to wondering if 0 probability events actually could occur, and the rest you know.

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