View RSS Feed

Memories of the 28th Century

Judging Probabilities

Rating: 2 votes, 5.00 average.
It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth. (From ďThe Adventure of the Beryl CoronetĒ by Arthur Conan Doyle)

Doyle presented an interesting idea, but I have to take exception to it; although it may just be a result of how we think of probability. There are some things that are possible that are so improbable that they might as well be impossible. It is easy to look at normal probability distributions in throwing dice and other common activities, but there are things that are not impossible but are so unlikely that they may be assigned a probability of zero (especially if we are rounding to nearest millionth), even though they may occur eventually. Probability is essentially a measure of past results. While there are mathematical descriptions of how probability can be expected in ideal situations, the real world is different. The mathematical ideals assume randomness, but randomness is, or may be, non-existent. There are many cases where the cause(s) of an event are not observable.

An example of this is when a radioactive atom emits a particle (the standard random event in Schrodingerís thought experiment). While we may not know why a particular atom emits at a given time, it is very likely that there is something, some precondition that makes the emission happen at a certain time. We have not observed that state, because we do not have equipment that is capable of providing us with a sufficiently detailed view of a nucleus. Such a lack of data can be expected to make our estimates of a probability inaccurate, but we canít tell how much so without detailed observations.

I am particularly interested in probabilities in regard to quantum theory and in regard to lottery drawings. There is a lot more information (relatively) available from lotteries than from some other sources. As with nuclear emissions, it appears that lottery drawings are not completely random. It is expected that any particular event is random and may be considerably different from the overall average, but in fact the behavior is not random in the short run, and there is a finite number of elements in the system that results in the numbers being selected, and the results. There are numbers that are preferred for variable periods of time. This is typical for most sorts of events; there are factors that affect the results that are not apparent. If we were able to learn all details of the system, then predicting results would be simple, but the level of detail is not available.

There are some things that can be predicted with a remarkable amount of success, while other things seem to be unpredictable. An example of the first sort of thing would be the rising of the Sun; it happens with great regularity, and has been doing so for a long time, and we can expect it to continue doing so for a very long time. Unpredictable things tend to be one time things or things that happen rarely. And there are events that are set up to be unpredictable, such as lottery drawings. Humans and their reactions to events are also rather unpredictable, but they are not designed to be unpredictable; that is simply how they are. Matters about which we can clearly observe all relevant elements and see and understand the interactions among the elements are usually more predictable than things with hidden, or unobservable, features.

Then thereís the matter of DNA, mapping relationships, etc. Y chromosomes can be followed for a fair number of generations, and there can be large numbers of people with the same Y chromosome. This is pretty solid evidence that they have a common male ancestor, but once mutations show up it becomes less certain. Like lotteries, the processes that may affect DNA are unobservable. We canít see what will result in a mutation, nor can we see how they are going to be sorted during meiosis. They are like the balls used to select lottery numbers; they move around, and then a set of them will be there in an embryo. We can make predictions based on our experience with outcomes, but the predictions would be more accurate and detailed, if we could see and understand all of the intermediary steps.

The same is true of many kinds of events; we could predict with more detail and accuracy, if only we could see and understand all of the details. We can tease a lot of information from the data, if we have enough data, but that doesnít replace detailed information.

To get back to Doyle, I have to disagree. If one eliminates the impossible, then one is left with the possible. But being possible does not mean that something actually exists. And there are possibilities that are so improbable that they are never seen. This makes applying the Many Worlds Interpretation to actual reality easier, because it means that there need not be an infinity of worlds, because some of the possibilities are not actualities. Sometimes I like to imagine those worlds sorted by probability, so that there will be a world where the very improbable goes, off there far from the probable worlds. There is an analogous situation for lotteries. While the usual suspects usually are drawn, sometimes a set of numbers comes up in which none had been drawn for months.

Once again, the conclusion is that following the trend really is the best policy, and it helps to realize that you will be completely wrong sometimes regardless of which way one might bet, and getting as much data as possible is also a good idea. If there is no data about something, then you canít expect to make any conclusions about it, but you wonít need to, because it is very unlikely to happen.