Tag Archives: randomness

The Semantics of Statistics

This has been brewing in my head for a while, and now that I’ve got a little time on Saturday night, I shall attempt to get my thoughts into a Word document before copying and pasting into WordPress at a later time.

I get quite annoyed when I read or hear people getting the wrong end of the stick when they talk about statistics and probability. Specifically, my beef is with the use of the word “random” in a very loose way. Of course, I don’t discount the possibility that I have been careless myself, but I can’t think of any examples. To illustrate this, it cropped up in a quite revealing conversation I had recently with a creationist.

I have laid out my position on this before, so I won’t go into too much depth. One trick that catches a lot of people out is the mixture of truth with untruth. In this particular conversation, I pointed out that some very good evolutionary science has had some very poor philosophy attached to it; but the problem that is created is that many of evolution’s best apologists fail to distinguish between them. What then happens is that those who lack the ability to discern between them are left with a choice to either reject or accept this mixture of ideas. Hence, you can often accept some poor thinking (as in the case of those who think evolution rules out God) or you can reject the good science (in the case of the creationist).

My observation, based on talking to a number of those who hold either of these views, is that there is a lack of understanding of what it means to be “random.” The phrases (or something equivalent to them) are that “evolution is a random process” or “genetic variations are random.”  Also, the word “chance” is used in this context though I find this so ambiguous, scientifically speaking, as to be almost useless.  The false interpretation, which seems to be relatively common, is that something which is random is indeterminate. What is true, however, is that it simply means something is unpredictable. Of course, if something is indeterminate, it will be unpredictable, but it is a logical fallacy to say that something which is unpredictable is necessarily indeterminate.

To demonstrate this, we need only consider chaotic motion. One example of this is the motion of a magnet suspended above several others. The motion is governed by a well understood interaction of electromagnetic forces and gravity. However, minor variations in the initial conditions will result in wide differences in the resultant motion. So by observing the initial conditions, one cannot practically measure to a sufficient degree of accuracy in order to be able to predict the motion.

Things get even more pronounced when you talk of quantum mechanics. As most people know, the problem of measurement can no longer become overcome even in theory. In the quantum world, probabilities rule. You no longer speak of a particles position, but rather of the probabilities of finding it in a given position. And if you do find its position precisely, there’s no way of knowing its momentum. For nigh on 100 years, there are have been competing ideas as to how to interpret this, ranging from the Copenhagen interpretation to the many worlds hypothesis.

It’s fascinating how probabilities change simply on the basis of the revelation of information. Anyone who has scratched their ends and eventually come to the right solution for the Monty Hall problem know that the crucial bit of information is that the host knows where the best prize is. Unlike the Monty Hall problem, Deal or No Deal has a host who is clueless as to where the prizes are located. The top prize is £250,000 and at the start of the game there is a 1 in 22 chance that the player has the box with them. But as the game progresses, and the £250,000 is not revealed, the probability increases. Nothing has physically changed about the box, only the information has changed.

Likewise, the last couple of weeks have seen exam results for Scottish Highers, A-Levels and GCSEs. All the papers have been marked and the exam results are known to the examiners. Yet to the students, with a sealed envelope in their hand their lack of knowledge of the contents means that the results could still go either way; they could get the grades they need or they might not. To them, the probability factor makes it indistinguishable from a crazy scenario whereby the results weren’t fixed until they opened the envelope.

I hope that made some kind of sense. I know it’s slightly disjointed. But I hope you found it interesting. Let me know what you think.

Book review: The Drunkard’s Walk by Leonard Mlodinow

I was first introduced to the writing of Leonard Mlodinow when I read Euclid’s Window as an undergraduate in mathematics. It was (and still is) one of the most delightful books on maths I have ever read. So it was with great anticipation that I looked forward to The Drunkard’s Walk.

The subtitle, How Randomness Rules Our Lives, is probably a more apt description for the book, as the question of the Drunkard’s Walk is not really discussed. If you are unaware of it, it relates to the probability that a drunk person, staggering around randomly will eventually return to where they started from. Given enough time, this probability tends towards 100% if they travel in two dimensions, but not if they travel in three dimensions. So if you’re looking for an explanation of why, this is not the book for you.

Rather, this is about probability and statistics. The mathematics in it is not particularly technical, although there are some concepts in it that seem to go against common sense. In fact, common sense is a running theme throughout the book; but only insofar as how unreliable common sense can be (here I am distinguishing between “common” sense and “good” sense). The book takes us on a wide and varied journey through history, sports, finance, gambling and medicine, amongst other things, looking at the way in which chance events are commonly understood, how they should be understood and getting to grips with the consequences of what happens when there is a gap between the two.

It is written in a highly engaging way, with enough humanity in it to keep the lay reader interested and just enough technical detail for the more mathematically minded to stay the course. There are, however, a few small points to pick up on. Mlodinow himself is a physicist, not a statistician . This results in a couple of explanations appearing a little muddled. For example, his explanation of why the probability of two independent events both occurring should be multiplied together may not be clear to someone without any maths knowledge beyond GCSE level.

The slightly bigger issue is in his philosophical interpretation of randomness. There is very little discussion over different interpretations of randomness and of what probabilistic measures mean. The conclusion that Mlodinow seems to reach is that it applies to unpredictability. However, in various places throughout the book, he seems to get this confused with purposelessness, which leads to a few unjustified conclusions on specific matters.

That said, this is a minor distraction in an otherwise excellent book.