Monday, June 23, 2008

The Problem with Probability

The way Probability is taught is really not conducive to remembering the material. It reinforces one of my biases towards engineers, and more so about mathematicians, that they may brilliant at what they do, yet they can make for extremely bad programmers.

Let me use some notations from probability as a discipline to demonstrate. I won't even mention concepts, except to say that each notation listed on a different line means a different thing, and things on the same line are different ways of saying the same thing:
p
p(x)
f(x)
P(X) P{X}
P(X=i) f(i) fx(i) P(X=x) fx(x)
P(X<i) F(i) Fx(i)
P(X>i)
P(X,Y)
P( a<x<b>X>a )
E[X]
Var(X)
C(a,b)
P(a,b)
and so on. In short it is a screwed up abuse of functional notation.

The common dogma is that probability is "a hard subject to teach and a hard subject to learn".
Maybe so, but this kind of gratuitously overloaded notation seems purposefully designed to make the subject difficult to communicate.

Think about the teacher writing a formulas with all those things up on a chalk board, then about the student taking the notes, then about the student trying to read the notes later and reason on the material. If telling the difference between these hand-written symbols is difficult, verbalizing formulas with them correctly is impossible:
"let's see, pee-of-ex paren ex less than ex equals eff sub ex paren ex"
could read P(X<x)=fx(x) or P(X<x)=Fx(x) etc. No matter that the experienced student may ferret out the correct meaning, sooner or later: the overloaded notation makes learning the topic, and reasoning in its language, needlessly difficult and hopelessly error-prone.

Here's a hint: it isn't productive to use upper case and lower case letters to distinguish between critical concepts throughout a subject, nor to abuse functional notation to mean something it doesn't mean anywhere else. A system of logic expressed in homonyms and synonyms is pedagogically deficient, regardless of the content, who is teaching it, or who is trying to learn.