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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

  1. Is this a known issue and where can I read more about this?
  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.

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  • $\begingroup$ "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? $\endgroup$
    – Jochen Glueck
    Dec 7, 2022 at 21:29
  • $\begingroup$ See the second paragraph in the question. A list of possible outcomes with their probabilities (while implicitly assuming a discrete distribution). $\endgroup$
    – Tommi
    Dec 8, 2022 at 7:39
  • $\begingroup$ 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. $\endgroup$
    – Jochen Glueck
    Dec 8, 2022 at 8:12
  • $\begingroup$ 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. $\endgroup$
    – Tommi
    Dec 8, 2022 at 13:29

2 Answers 2

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At the initial stages it is always "drill to kill". Literally, there is no interesting issue there. Just tell them to write the answers in full sentences always starting with "The quantity X can possibly equal to " when the question is "Find the probability distribution of the quantity X" and reduce points for not writing this sentence. Then tell them to make a full value versus probability table and, again, insist on the prescribed format. Those are idiotic requirements when the person knows perfectly well what he or she is doing, but if they don't, they help both them and you: if somebody writes in good faith "The sum of the numbers on two dice can possibly equal 1,2,3,4,5 or 6, you can immediately discuss just that particular sentence with them without trying to figure out what and where went wrong in the solution written in the free style format.

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  • $\begingroup$ Having this as a part of written assignment might work. This answer does imply the problem is simply forgetting the matter, or not seeing it in the first place, rather than any conceptual issue. Maybe it is. $\endgroup$
    – Tommi
    Dec 10, 2022 at 7:47
  • $\begingroup$ @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. $\endgroup$
    – fedja
    Dec 10, 2022 at 11:29
  • $\begingroup$ The very simplicity of it is why I was so surprised the students had not picked it up. $\endgroup$
    – Tommi
    Dec 11, 2022 at 9:05
  • $\begingroup$ @Tommi I completely agree with you. I told you what I am doing in such cases and it works more often than not for me (like I taught the students how to find the shortest path from A to B in a graph this way and to prove that it is the shortest one, indeed, in Discrete Math. and it was noticeably easier for them than my previous idea of explaining the general concept of the Bellman function/dynamic programming first). The requirement for a certain answer/proof structure facilitates both the correct thinking process and location of errors, IMHO. It just shouldn't be applied fanatically :-) $\endgroup$
    – fedja
    Dec 11, 2022 at 13:42
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The only way I have ever had success with students understanding what a probability distribution is by taking data, building a frequency distribution, and then asking questions about what we should expect if we chose an individual at random, given that frequency distribution.

For example: if you have 30 students in your class, ask them all a hopefully bland, non-intrusive question. Maybe "how long is your commute to school" or "what is the fourth digit of your phone number"? You can probably come up with better examples.

Once we have the frequency distribution, we can talk about it approximating a probability distribution, and hopefully make connections as to why its shape makes sense and what that means for expectations.

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