Why is probability a tough Subject, even for students who are very good at math?

An answer that I was trying to answer on Quora, but was prevented from doing so by a technical fault: Why is probability a tough Subject, even for students who are very good at math?

Because probability as it manifests itself in real life is surprisingly deep and difficult concept. As a knowledgeable colleague once explained to me,

There are at least four distinct interpretations of probability:

  • objective Bayesianism
  • subjective Bayesianism,
  • a propensity theory
  • a frequency theory

along with various pluralists positions.

Unless you work in artificial situations with, say, perfect dice, these differences, which I imagine most school teachers are unaware of, will confuse one’s teaching.

Odds in horse races provide a very good illustration of probabilities. Are they

  • The unique propensity of a horse in that precise situation to win.
  • The limiting frequency in some long series of events.
  • A measure of subjective expectation, reflected in betting behaviour.
  • An objective measure of the expectations of a rational agent given certain information.

The question is: what should we tell to children? At what age they become able to distinguish between “subjective” and “objective”?”