Friday, 2 July 2010

Probabilities of Averages

Consider something that has a very low probability of happening, like rolling the sum of 100 when rolling one-hundred six-sided dice. Since there is only one combination with this sum (i.e. all 100 dice show one pip), the probability of it happening is one in 6100. If the sum we are interested in is instead 101, then one die must end up with two pips showing. This could be any of the dice, so even though the probability of this happening is actually still very small, it is still a hundred times greater than rolling the sum of 100.

Now, when rolling only one die the average roll (over an infinite number of rolls) is 3,5. Of course when rolling only one die the probability is also equal for all results. If we increase the number of dice with one, the average sum will be 7 (3,5*2). If we name the dice d1 and d2 we can describe a roll as [d1, d2]. Rolling the sum of 7 can then be achieved by rolling any of the following combinations: [1, 6], [2, 5], [3, 4], [4, 3], [5, 2], [6, 1]. Each combination has the same probability of 1/36, but when only considering the sum of 7 the probability is as much as 1/6. A higher or lower sum than average will decrease the number of available combinations and thus the probability for the sum, though the probability for each combination will still be the same. At the extreme ends are the sums of 2 ([1, 1]) and 12 ([6, 6]), with one combination each.

With one hundred dice the average roll would be 350 (3,5*100), and just as we saw earlier the probabilities are not spread out evenly over the different sums. This means that a sum of 350 has a bigger probability of happening than other sums. If we would use the notation from above we could describe a roll as [d1, d2,..., dn]. We would then see that just as in the case of two dice, the number of combinations for a sum decreases as we stray away from the average sum, so that it is very likely that we end up with a sum in the vicinity of 350.

So what do I want to say with this? Perhaps nothing more than that I like probabilities. This reasoning can also be applied to a lot of things around us though. The average combination of events around us will lead to something that we can usually predict but the result has a probability of being something else. In fact, according to chaos theory the only reason the laws of physics can be reasonably relied on is that they include so many die rolls that it is too likely for them to not stray from the average.

I would like to point something out also about everyday coincidences. When something strange with a low probability happens it might not be as strange as we would like to think. In the period when one unlikely event happened, millions of likely events will have happened as well. There is also the fact that it might be likely that something unlikely is to happen even though a particular unlikely event has a very low probability. An example would be to generate a random number between one and a billion, each number has a very small probability but it is very likely that we will get one of those numbers still.

It is getting more and more probable that this article is getting long-winded so it is very likely that I will end it here.

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