# How to teach the concept of probability distribution?

I observed that my students do not understand what a probability distribution is.

We do not treat probability axiomatically on the course, so the required level of understanding is knowing all the possible outcomes and their probability, which is good enough for discrete distributions. We do not pay too much attention as to what this might mean with continuous distributions; I would be happy if the students understood the meaning for discrete ones.

When asked about the probability distribution of, say, the sum of two dice, I am equally likely to get the expected value or the probability of a single outcome, or even the probability of getting a given result on both dice, than I am to get what I would think of as the probability distribution.

I have had students calculate some probability distributions and things based on them, but this is not a huge theme on the course. I have also given them the meaning of the term "probability distribution". We have seen different representations of probability distributions, such as trees, graphs and the area model. It is unclear whether I will cover probability distribution as a concept next year and how thoroughly.

Questions

2. Any good ideas as to how to work with probability distribution as a theme?

Context

Teacher education of future teachers at grades 5-10 in Norway (mostly grades 8-10 and thereby lower secondary education). The students are second year university students with 40 ECTS credits in mathematics didactics, but no previous university level education in probability. Many have only practical (low-level) mathematics from videregående (upper secondary school) and grades might not be very good.

Literature

This year we used Alfa as the main book, supplemented by QED 5-10 -books. Next year we are likely to have QED as the primary textbook. Hva er sjansen for det? does not cover the subject. Other books, preferably in Norwegian or other Scandinavian language, are possible.

• "I have also given them the meaning of the term 'probability distribution'." Could you elaborate a bit? For instance, did you give them a precise definition of the notion "probability distribution", and if yes, which one? Dec 7, 2022 at 21:29
• See the second paragraph in the question. A list of possible outcomes with their probabilities (while implicitly assuming a discrete distribution). Dec 8, 2022 at 7:39
• Hmm, yes - my point rather was the difference between wording it as (i) "We now discuss probablility distributions. Knowing the probability distribution means knowing all the possible outcomes and their probability" (as in the second paragraph) and (ii) "The probability distribution is the list of all outcomes together with their probabilities." Back then as a student I would have been quite comfortable with (ii), but would have been irritated by (i) - so I was wondering whether part of the issue might be how explicitly one tells the students the definition. Dec 8, 2022 at 8:12
• Hard to remember such details, but feel free to answer along those lines. Most of these students are not at the rigorous level of mathematics. Dec 8, 2022 at 13:29

• @Tommi Frankly, I find the concept involved absolutely straightforward because it is purely algorithmic. It is at the level of "to differentiate $x^p$, pull $p$ to the front and reduce it by $1$ in the power", albeit with more computations. In such cases, repeating the formal rule in writing (or orally) every time before carrying out the related manipulations should quickly lead to the memorization of the rule and making no mistakes in its overall application to simple examples. Once that is achieved, one can try to discuss subtler issues, if you see any here. Dec 10, 2022 at 11:29