I am planning to become a math high-school teacher and have the following question:

What Probability Theory should I base myself on to teach probabilities to students ?

The classical approach is via the Kolmogorov axioms. The second one would be the Probability Theory via the Cox theorem, Probability Theory whose foundations are described in Jaynes' book Probability Theory: the Logic of Science.

These two SE questions (this and this) summarize the antagonism between the two approaches.

The thing is that I am much more confortable with the second approach, and (since I know almost nothing about Measure Theory) I am unconfortable with the classical Kolmogorov approach.

Yet, I am afraid to teach the second approach (the one taught by Jaynes) because:

  • I am afraid to teach the "wrong math" to students (see why below).
  • even if it is no "wrong math", I am afraid that an inspector comes along and punishes me for teaching what he thinks are wrong math.

I am totally unable to say whether Cox theorem derivations are weaker than the Kolmogorov approach, whether they are but it doesn't quite matter to teach Cox-based Probability Theory.

Yet from what I have understood, countable additivity is not ensured by Cox-based Probability Theory even though this paper seems to claim that both approaches are equivalent.

I am aware that probably the best thing I could do is to teach the Probability Theory I am most confortable with and prepare to elaborate on why I chose it to the hypothetical inspector ? However, could there be something wrong with that ?

Thanks in advance.

PS: I am of course not going to teach the whole theory (be it Kolmogorov or the other one) to students, but I want to know what should I derive from what (and what I can use to prove these derivations).

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    $\begingroup$ This is high school in the US? I think both of these are too advanced for those students. $\endgroup$ Commented Jan 21 at 18:37
  • 2
    $\begingroup$ No, not in the US. But depending on the Probability Theory foundation we choose, derivation of basic results are different. E.g. in Jayne's approach, $p(A)+p(B)-p(A\cap B) = p(A\cup B)$ is derived from Bayes' rule (aka. conditional probability rule) and $p(A)+p(A^c) = 1$. On the other hand, $p(A)+p(B)-p(A\cap B) = p(A\cup B)$ is derived, in Kolmogorov approach, from the three axioms of Kolmogorov. $\endgroup$
    – niobium
    Commented Jan 21 at 18:58
  • 14
    $\begingroup$ Where is this being taught? I would expect an axiomatic approach to probability to come after courses in both calculus and something like "introduction to proofs" (often a discrete mathematics course in the US, but analysis wouldn't hurt, either). This seems way over the heads of the high schoolers I know. $\endgroup$
    – Xander Henderson
    Commented Jan 22 at 15:30

7 Answers 7


I would recommend avoiding foundational issues when teaching probability at a low level. At the high school level one mostly deals with finite probability spaces and the normal distribution. The Kolmogorov axioms, which in any case you will probably not teach fully to (normal) high school students, are the safest approach. Kolmogorov explicitly avoided justifying his axioms very much beyond pragmatic considerations - they work and they encompass the important examples known then (why won't you teach the Kolmogorov axioms? The key one, the continuity axiom, is about passing to infinite limits ...)

While relative frequency and area do not provide an adequate conceptual basis for the foundations of a probabilistic theory that encompasses processes such as Brownian motion, they work just fine at the introductory level.

I have sympathy with the de Finetti/Cox/Jaynes/etc style approaches to foundations (by the way, the three mentioned are different), but these are hard to find written rigorously and their practical implications are mainly relevant when trying to formalize how one does inference. If you look at Cox's nice book you won't find much directly useful for teaching at the high school level (although it might change how you think about what you teach). For a teacher better to read Polya's book on plausible inference (I can't remember the title) and follow the line he advanced there. An approach like that of de Finetti starts with the theorem on exchangeable sequences - but this is quite technical even at the introductory university level - the idea of exchangeability as more fundamental than independence and perhaps as justifying (via de Finetti's theorem) in practice the assumption of independence is quite appealing, and it might make sense to mention it in an introductory university level course aimed at mathematics students, but it will be totally lost on high school students who in general will have considerable difficulty to even understand the definition of independence, much less that it is a defined concept, rather than a fact of nature.

I like Jaynes's book, but it is intentionally provocative and argumentative and also is itself not unproblematic. Many professional probabilists and statisticians disagree strongly with his view point, and almost all will quibble with some of his more extreme claims. In the high school setting one wants students to understand something far more basic than what Jaynes is arguing about - namely that while deductive inference tells us that just because A implies B there is no need that B implies A, Bayes tells us that when A implies B and we observe B we gain more confidence in A. This point, which is often lost on professional mathematicians, is very basic, but it does not depend on whether one follows Jaynes or Kolmogorov or someone else.

  1. I think you should (and likely will have to) use the assigned text and approach. It's incredibly unlikely you will just derive some new approach. That's not how high school teaching works.

  2. And on top of everything else, you've got other things to concentrate on, not just curriculum design, but pedagogy, behavior management, etc. that will be higher priorities.

  3. And it's not even clear that you would be assigned a probability course, as a new teacher. You might be teaching more traditional courses. For that matter, often the modern approach is to slide a little probability (and stats) into the basic courses themselves. In which case, the content will be very clearly constrained.

  4. FWIW, even at the "not the whole theory", your approach seems way TOO HARD for an initial approach. You should instead do VERY basic frequentist stuff. And the approach is not "what it's derived from", but descriptive: "Here's a permutation and the formula for it." Rigorous justifications can wait for college and will only be needed for math majors...not the vast amount of STEM students in hard and soft sciences and in engineering.

  • $\begingroup$ Since this is not the US, there indeed should be an assigned text and approach. $\endgroup$
    – Rusty Core
    Commented Jan 22 at 6:04

I'm not familiar with Cox approach at all, so I cannot provide a qualified comparison, but I find the Kolmogorov's axioms pretty easy to comprehend and use and I'll try to explain why.

No matter what approach you take, you should probably start with the finite discrete setting when you have just finitely many possible outcomes $\omega_1,\dots,\omega_n$ forming the set of elementary events $\Omega$ with some probabilities $p_j$ assigned to them, so that the probability of a compound event $E$ is just $\sum_{j:\omega_j\in E}p_j$. You can then teach almost all relevant concepts up to independence and conditioning in this simple setting and I wouldn't move any further until this part is mastered entirely.

The second useful initial setting to cover is the "geometric" probability, when you have a random point distributed in some domain in $\mathbb R^n$ (you start teaching it for $n=1$ and $2$, of course) according to some nice continuous density and the students learn to find the probability of an event by figuring out what part of the domain corresponds to it and integrating the density over that part. Here you may have a gentle nab at the countable additivity already (like asking "what is a probability that a random variable uniformly distributed on $[0,1]$ is rational"). Again, before that part is mastered, I would not proceed to any general axiomatic setup.

Kolmogorov's approach then just generalizes these settings in (IMHO) the most direct way. You still have the set $\Omega$ of elementary events $\omega$ and the compound events are just the "reasonable" (measurable) subsets of $\Omega$. You don't need to know what exactly "measurable" means, only to understand that all reasonable events (like $X\in[a,b]$ for random variables $X$) are measurable and that you can perform all finite set-theoretic operations and countable unions and intersections without destroying measurability. Also, you can postulate the countable additivity of the probability. It, of course, raises the question why at least one non-trivial (not discrete and finite) probability space exists, but I doubt your students will care of it any more than about why the set of real numbers exists and I don't recommend that you care about that too much yourself in the beginning. At this point it is important to emphasize that not everything is an event and that one can express a lot of things using only countable operations, but not everything.

Once that is understood, you can learn/teach pretty much anything you want, just sweeping under the rug the related existence questions (like why can one construct a probability space with an infinite sequence of independent random variables uniformly distributed on $[0,1]$, etc.) but not the questions why something is an event.

That is a brief summary of the Kolmogorov's approach as I see it. The only essential thing that I didn't mention in the above outline is the conditional expectation with respect to a $\sigma$-subfield of the original $\sigma$-field. However, that concept is not hard too, plus I doubt it very-very much that you would ever have an opportunity to reach that far in the high school probability course.

As to the "inspectors punishing teachers for teaching wrong math", yeah, that is always a possibility.

  • $\begingroup$ lol i agree! btw what is the Cox view of probability? $\endgroup$
    – user123945
    Commented Jan 22 at 8:02
  • $\begingroup$ @user123945 a view according to which "probability is a measure of plausibility of a statement based on specified information". $\endgroup$
    – niobium
    Commented Jan 22 at 10:05

Should I choose Cox-based Jaynes' approach or Kolmogorov approach to base myself on to teach probabilities to high-school students?

My answer is: No, you should not do that.

These theoretical foundational issues are way outside any high school curriculum, and would frankly be a waste of time for you to spend working on.

There should be a received curriculum and an established textbook for the course. The authors have likely spent years or decades refining the best scaffolding for students of the given class level. The probability concepts in that book are almost surely predicated on an equal-likelihood model and the frequentist interpretation.

Read that book and follow it closely (sequencing, terminology, notation, exercises, and problem sets). You will have your hands full interpreting the book for students, establishing a schedule, writing and grading assignments and tests, answering questions, behavioral issues and grading disputes, etc., etc. Don't add outside work that doesn't benefit that, especially when you're a new teacher.

Side piece of advice for any teacher: You're almost certainly assigned to a course in statistics. Don't turn it into a course essentially in probability.

I am aware that probably the best thing I could do is to teach the Probability Theory I am most confortable with and prepare to elaborate on why I chose it to the hypothetical inspector?

If this a thing where you are, then I'd guess that the most likely risk from any inspector would be if they see that you are wildly off-book and off-curriculum in the classroom; for example, if you're spending time on foundational probability theory that is outside the book and inappropriate for that class level.


"I am planning to become a math high-school teacher"

So it seems to me that you expect too much from high school students. I made the same mistake, the true is the vast majority of them will not understand (or even care for) a derivation of the basic probability results. Be grateful if they show interest to use these results to solve easy problems.

So what is truly important at this level is the pedagogy, You have to engage them to the themes with interesting examples, remember that at this age the life is not around maths. In my case, I try to center my class in showing that solving math problems can be very useful in exercising the brain and learn to deal with stressing situations.


In Economics of past decades, an analogous issue arose as regards teaching "the neoclassical paradigm" versus teaching the "Keynsian paradigm".

In a pretty realistic approach prominent neo-classical economist Barro in his textbook wrote that Keynsian economics is an advanced topic, and students should first be exposed to the neo-classical paradigm, as a base to pivot afterwards.

The same holds here: "Kolmogorov axioms" align with the most basic intuition we can have about chance and probabilities, while the Cox/Jayne approach, even though Jayne did build his theory from scratch, is an advanced stage.

I also mention that in his main book Jayne finds himself in agreement with the Kolmogorov approach on many aspects, and he does not hesitate to explicitly say so. For example he writes on p. 49:

In Appendix A we describe the Kolmogorov system and show that, for all practical purposes, the four axioms concerning his probability measure, first stated arbitrarily (for which Kolmogorov has been criticized), have all been derived in this chapter as necessary to meet our consistency requirements. As a result, we shall find ourselves defending Kolmogorov against his critics on many technical points.

And then on p. 50 he concludes:

Our system of probability, however, differs conceptually from that of Kolmogorov in that we do not interpret propositions in terms of sets, but we do interpret probability distributions as carriers of incomplete information. Partly as a result, our system has analytical resources not present at all in the Kolmogorov system. This enables us to formulate and solve many problems – particularly the so-called ‘ill posed’ problems and ‘generalized inverse’ problems – that would be considered outside the scope of probability theory according to the Kolmogorov system. These problems are just the ones of greatest interest in current applications.

Namely: advanced stage.


How probable is it that a $1$ will appear when you throw a die?

How probable is it that there was life on Mars a billion years ago?

For the first question, one can say $1/6$ because it happens one-sixth of the time.

For the second question, if the answer is $1/6,$ that does not mean it happens one-sixth of the time.

You can use the axioms of mathematical probability to model relative frequency, as in the first example, or degree of belief, as in the second (that's the Cox–Jaynes approach, usually called "Bayesian" although Bayes himself probably had nothing to do with it).

Either way, you'll use the Kolmogorov axioms.

  • A probability space, or "sample space", is a set of all possible "outcomes."
  • An "event" is a (measurable) subset of the sample space.
  • A probability measure is a function assigning probabilities to events in a way that is (countably) additive and assigns measure $1$ to the whole space.
  • A random variable is a (measurable) function whose domain is the sample space.

You'll use that regardless of whether you take a frequentist or a Bayesian approach.

And I probably wouldn't talk about frequentism versus Bayesianism in a high-school course unless the topic somehow comes up when I didn't initially intend it to (e.g. a student mentions it or it becomes relevant to an particular exercise).


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