Bernoulli Trials and Tribulations

This one feels relatively minor to us. It is, however, a clear own goal from the VCAA, and it is one that has annoyed many Mathematical Methods teachers. So, as a public service, we’re offering a place for teachers to bitch about it.*

One of the standard topics in Methods is the binomial distribution: the probabilities you get when repeatedly performing a hit-or-miss trial. Binomial probability was once a valuable and elegant VCE topic, before it was destroyed by CAS. That, however, is a story is for another time; here, we have smaller fish to fry.

The hits-or-misses of a Binomial distribution are sometimes called Bernoulli trials, and this is how they are referred to in VCE. That is just jargon, and it doesn’t strike us as particularly useful jargon, but it’s ok.** There is also what is referred to as the Bernoulli distribution, where the hit-or-miss is performed exactly once. That is, the Bernoulli distribution is just the n = 1 case of the binomial distribution. Again, just jargon, and close to useless jargon, but still sort of ok. Except it’s not ok.

Neither the VCE study design nor, we’re guessing, any of the VCE textbooks, makes any reference to the Bernoulli distribution. Which is why the special, Plague Year formula sheet listing the Bernoulli distribution has caused such confusion and annoyance:

Now, to be fair, the VCAA were trying to be helpful. It’s a crazy year, with big adjustments on the run, and the formula sheet*** was heavily adapted for the pruned syllabus. But still, why would one think to add a distribution, even a gratuitous one? What the Hell were they thinking?

Does it really matter? Well, yes. If “Bernoulli distribution” is a thing, then students must be prepared for that thing to appear in exam questions; they must be familiar with that jargon. But then, a few weeks after the Plague Year formula sheet appeared, schools were alerted and VCAA’s Plague Year FAQ sheet**** was updated:

This very wordy weaseling is VCAA-speak for “We stuffed up but, in line with long-standing VCAA policy, we refuse to acknowledge we stuffed up”. The story of the big-name teachers who failed to have this issue addressed, and of the little-name teacher who succeeded, is also very interesting. But, it is not our story to tell.

 

*) We extend our standard apology to all precious statisticians for our language.

**) Not close to ok is the studied and foot-shooting refusal of the VCAA and textbooks to use the standard and very useful notation q = 1 – p.

***) Why on Earth do the exams have a formula sheet?

****) The most frequently asked question is, “Why do you guys keep stuffing up?”, but VCAA haven’t gotten around to answering that one yet.

43 Replies to “Bernoulli Trials and Tribulations”

  1. I’ll answer ****) for you, marty. It’ll save VCAA the trouble – as they’re a bit too busy nitpicking dx’s, brackets and notation to give a few seconds of their precious time.

    Q: “Why do you guys keep stuffing up?”
    A: “Because we’re precious dickheads who refuse to admit our mistakes. We’re always correct, and students/teachers who make the slightest of errors will be penalised and talked to sternly in the exam assessor’s report (which will take its own sweet time getting published, because as we said before, we’re very busy finding things to penalise students for). Now get back to your meaningless blog, because we don’t care about what a bunch of annoyed teachers think of us. All hail our overlord, DLL”.

    There you go!

  2. I’m confused. Are VCAA just saying that binomial distributions with n = 1 are examinable but the phrase ‘Bernoulli distribution’ will not appear in an exam?

    1. SRK, it’s not totally clear, but that’s my take.

      “Binomial distributions with n = 1” would presumably have to be examinable, since binomial is examinable. So, I can’t see that the question is anything but “Is the jargon ‘Bernoulli distribution’ examinable?”, and I take the FAQ to be answering with a definitive “No”.

      1. They also state that Bernoulli trials are examinable.

        I wonder how many students / teachers are going to worry if they’re missing some sort of subtle distinction between (i) a Bernoulli trial, (ii) the Bernoulli distribution, and (iii) the binomial distribution with n = 1.

        1. Very good point, and I’m sure that’s part of the annoyance. Above I noted “Bernoulli distribution” is close to useless jargon, but you are also correct that, in conjunction with “Bernoulli trial”, it actively invites confusion.

        2. Hi SRK (and Marty). We’re going to disagree on this point.

          There’s no distinction between (ii) and (iii). Once the definition is given I don’t think any competent Methods teacher could see a distinction.

          So it boils down to whether there could be confusion between Bernoulli *trial* and Bernoulli *distribution*. A *trial* is an *experiment*, a *distribution* describes the *outcomes of the experiment*. How can there be confusion? Part of teaching mathematics is to teach precision in language.

          OK, there probably will be confusion, but there shouldn’t be. Do things like ‘normal distribution’ and ‘standard normal distribution’ cause confusion …? How about ‘mutually exclusive’ and ‘independent’? ‘Equilibrium’ and ‘static equilibrium’. etc.

          There will always be confusion if people don’t read things carefully. Although admittedly, VCAA has done its best to (again) sow confusion.

          1. JF, normally I would agree with you. But, in this case “Bernoulli” has suddenly and clumsily and uncertainly gone from referring to one idea in probability to two ideas. I wouldn’t bash any Methods teacher/student too hard if some hypothetical (or actual) question confused them.

  3. I’m familiar with Little-Name’s story. It’s probably not quite as interesting as Marty’s ‘cliff-hanger’ suggests. The way I hear it:

    Little-Name sent an email to VCAA querying the addition, wasn’t happy with VCAA’s reply and followed up with a few more emails. Along the way, a big-name teacher 60.1282° N, 18.6435° E gave some much appreciated support, and there were very senior people at VCAA who were willing to listen to Little-Name and ensure appropriate action was taken.

    One can speculate on the weasel-worded (everything after the single word ‘No.’) update of the FAQ page. My theory is that the exams including the formula sheet are already printed and VCAA can’t just pulp and re-print them. An explicit admission of error would require this action. So an updated FAQ page and, apparently, a clarifying memo sent to all schools is the very best outcome that could reasonably be expected.

    A happy consequential outcome might also be an increased oversight and scrutiny of the maths exams and the Examination Reports.

    In reply to SRK: Obviously Binomial with n = 1 is examinable. But it’s the *name* of this special case distribution – The Bernoulli Distribution – that is *not* examinable. The *name* is jargon that does not appear in the Study Design, any VCE textbook, any past VCAA exam, or any present or past commercial trial exam. There is no way a student would see *Bernoulli Distribution* and understand that it meant Binomial with n= 1 unless that jargon was explicitly taught. And obviously (and quite reasonably) it won’t have been taught. *That’s* what makes it’s inclusion on the Formula Sheet stupid, irresponsible and wrong.

    It follows from all the bullshit after ‘No’ in the Updated FAQ, that the distribution of a sample proportion, deleted from the Adjusted Study Design, is nevertheless examinable because it has a “natural connection” to the Binomial distribution … The phrase “natural connection” is a wonderful example of VCAA-speak. The phrase can used to justify the inclusion of almost anything on the exams that is not explicitly stated in the Study Design. Which is why it’s so important to have a Study design that is clear, explicit and *trustworthy*. What’s the probability that the current so-called ‘Expert’ Panel will produce such a document? Hint: It follows a Bernoulli Distribution with p = 0.

    1. Terry, the comments *are* pedantic. But, as you seem to agree, it is justified pedantry. It’s justified because teachers and students must cope with VCAA’s pedantry. And, the problem is not just that the VCAA are pedants, which is awful enough, but that they are incompetent pedants.

    2. This is exactly the point.

      I’ve thankfully been given the year off teaching Methods but I have seen how this change to the formula sheet has shaken a few proverbial trees:

      1. No teacher I spoke to was aware of what the Bernoulli distribution specifically was and some went quickly looking worked examples.
      2. Since this was never on the formula sheet (which is itself pretty redundant for paper 2 since bound references are permitted), a few predictions of how (not if, at this stage) it was going to be examined on paper 1 started flying around.
      3. The FAQs didn’t really help that much as a few years ago the “consideration of normals is not required” line which was contradicted by a “perpendicular line to the tangent at point …” question on the subsequent exam just confused people even more.

      So Terry’s point is very, very relevant.

      1. Hi RF.

        Re: 1. As Dirty Harry once said, those teachers would have been shit outta luck trying to find examples.

        Re: 2. The only explanation (apart from incompetence and stupidity) why the Bernoulli Distribution would be ADDED to the formula sheet is if it was examined on the paper. I’ll admit I went with ‘examined on paper 1’ instead of the much more likely ‘incompetence and stupidity’ explanation. And yes, I wrote some sample questions to give to my students after telling them that Bernoulli distribution simply meant Binomial with n = 1. My students were actually more interested in what a Bernoulli distribution could possibly be used for than anything else!

        Re: 3. Marty’s too nice …

        1. So… IF a question on Bernoulli distribution does appear on a VCAA exam this year, despite the FAQs saying it is NOT examinable…

          Yeah, we know how this story probably ends.

          1. RF, let me modify your comment slightly:

            IF a question *using the phrase* Bernoulli distribution does appear on a VCAA exam this year, despite the FAQs saying it is NOT examinable…

            Yeah, we know how this story probably ends.

            RF, if the above does happen I don’t know how the story will end but I do know how the middle part will go, and the ending could turn out to be a happy one.

            1. Happy for whom?

              I try not to engage in speculation unless it involves horses, but VCAA kind of invites it with this sort of action.

              The secondary issue is that IF such a question happens to appear, it will be forever before an examiners report appears and even then, if students somehow do well (multiple choice perhaps) the report will be less than helpful in planning for 2021.

              1. I’m hoping it would be a happy ending for every ‘no friend of VCAA’ …

                I’ll be surprised if such a question does appear because I like to believe the exams and reports will come under much closer internal scrutiny this year … And it’s even possible the reports might come at some time late/early term 1/term 2.

                1. Are you a gambling man, JF?

                  I will give you generous odds that we do not see the examiners report before May 1st, 2021 and that the paper itself is not published online before March 1st, 2021.

                  (Insert random wise crack about this being based on a Poisson distribution)

                  1. It certainly feels like getting kicked in the head by a Prussian horse …

                    OK, I’m a sucker. I think there’s a new(ish) broom that will sweep VCAA clean. I’ll take both bets, we just have to agree on the stakes ….

                    (Let this blog be the witness – I can feel the excitement building already).

                    1. I have a signed copy of “A dingo ate my math book” by Polster and Ross…

                      …actually, I’ll think of something else, I don’t want to risk losing the inscription. I’ll think of something else to offer.

                      But if you have inside information, I consider your advantage a bit unfair, but will nonetheless acknowledge your victory and pay any winnings due, because I’m not an online bookmaker.

                    2. I don’t think I have inside info, but maybe I have. I’m not sure. Let’s call it a gentleman’s bet (I doubt there could be a more INappropriate name given at least one of the ‘combatants’, but nevertheless …) and the winner gets bragging rights.

    3. Indeed. It’s a damning indictment of VCAA that the Study Design is not similarly pedantic. Because you’re 100% correct, TM – teachers *do* have only the written word to guide them (and have to regularly look to (dodgy) exam questions and (dodgy) reports to join the dots). We all clearly need a greater understanding of “natural connection” …

      On this theme, how many teachers think that integrals of the form \displaystyle \int \frac{f'(x)}{f(x)} dx (and f(x) is NOT a linear function) are part of the *Maths Methods* course? It might surprise some teachers to know that VCAA and its toadies think that they are (although VCAA have qualified this by saying it was highly unlikely such integrals would appear on the exam). And yet the formula is not mentioned in the Study Design or the formula sheet. What does this tell us?

      1. JF, I had no idea that integrals of the form you stated are part of the Methods course. I’ve known all along that it’s part of Specialist, using a u-substitution.
        No wonder – I got a shock when reading through the 2020 MAV Methods Trial Exam 1 and saw a question that would have required this!

        1. Steve,

          The author of 2020 MAV methods exam has very strong expertise in writing VCAA spesh exams.

          Maybe this person inevitably used some of the “basic ideas” from Spesh when writing questions up.

          JF,

          Big name teacher “60.1282° N, 18.6435° E” is a great mathematician and must be appreciated by all of us. Once he has created so many great resources as what you did in the past (which you probably couldn’t find anywhere else nowadays)
          I really hope his great ideas and sophisticated questions come back to actual VCAA exam(s) one day, instead of our students tolerating with the lengthy, wordy, error-not-free papers.

          Bernoulli distribution? All in all, VCAA’s actions this year overcomplicated things. You are totally correct: X~Bi(n=1, p=p), that’s it.

          Some relevant MCQ May look like:
          Which of the following is a Bernoulli random variable?
          A. The length of waiting time at KFC drive-in
          B. The number obtained uppermost by rolling a fair die
          C. The number of heads obtained when flipping a coin twice
          D. The number of heads obtained when flipping a coin three times
          E. Whether VCAA can produce an error free methods exam 2 in 2020 (LOL)

          Marty,
          I agree with JF, you are indeed too nice and lovey-dovey this time.

          1. The MAV writer(s) have form for doing this (see the 2011 MAV Exam 1 Q1(b) – written by the same person – for an even ‘worse’ example) despite repeated arguments that it’s NOT part of the course.

            It is no accident caused by *ahem* ‘strong expertise’ (pardon me while I choke with incredulous laughter). This MAV writer has wilfully and consistently argued that \displaystyle \int \frac{f'(x)}{f(x)} dx IS part of the Methods course – VCAA has ‘supported’ this view but then distanced itself from this view by saying it was highly unlikely such integrals would ever appear on the exam. Typical VCAA – unwilling to say yes or no to a simple ‘yes or no’ question.

            I’ve heard that the only arguments the writer can offer is that:

            1) they see their question as being of the same ilk as finding \displaystyle \int_{4}^{5} \frac{2}{2x-1} dx (VCAA 2014 Exam 1 Q2), and therefore

            2) knowing f'(x)/f(x) is an advantage.

            3) \displaystyle \int \frac{1}{x} dx is on the formula sheet and that’s a sufficient prompt for students to know \displaystyle \int \frac{f'(x)}{f(x)} dx.

            And a correction: The writer has extensive experience watching other people with expertise write Specialist exams. This does not equate to the writer themselves having any expertise (let alone *strong* expertise) except in the vicarious sense. And it’s a poor excuse for the writer to be continually putting Specialist questions on a Methods exam. Personally, I think at best the writer is simply trying to ‘show off’ – I suppose it impresses the gullible and the green horns. At worst, ….

            I believe a blog on commercial exams is imminent so I’ll refrain from further comment on the numerous other issues that particular exam has and the superficial mathematical understanding it showcases. Which the MAV endorses.

            1. John, and everyone, any exam question not covered by the study design (which may be a judgment call) is an error. Please make comments to that effect on an Error List post and I’ll look carefully, to see it should be added.

              1. Will do so right now marty. Will head over to the MEL post and mention these two exams (MAV 2020 Exam 1 and MAV 2011 Exam 1 [courtesy of JF]).
                Looking forward to see where the discussion heads there.

            2. JF, I just looked up the specific question in the exam you stated – and yes I see it happening again! It’s been a while since I’ve laid eyes on that particular exam so it’s escaped my memory, but you’re indeed correct.

              Fortunately, the Specialist exams (at least Exam 1) aren’t flawed like that, in large part to the astute author 😉

        2. Steve, I can imagine your shock. Especially since the reality is that they’re NOT part of the Methods course, despite idiots who argue the contrary (see my comment further down for the idiot arguments) and despite VCAA’s (typical) wishy-washy non-committal response to a simple ‘yes or no’ question. Have the pitchforks and rope ready if a question like this appears on a VCAA Methods Exam 1 …

          And its idiots like these that are probably on the ‘Expert Panel’ to determine what VCE Mathematics will look like for the next couple of years …

      2. Integration by parts is not in the methods study design, but how often have we seen “Part (a): differentiate …”, “Part (b), hence, find an anti-derivative of…” questions? (A lot)

        So the examiners ARE testing a skill which is not in the study design but defend doing so because the exam question was *guided*.

        Yep. Here we go again… (probably)

        1. RF, I don’t think that’s really the same. But why these cretins just don’t include substitution/parts in the Methods/Specialist syllabi is beyond me.

  4. Hi,

    Seems like a lot of words about an ACE …

    so No doubt there will be a question on the Bernoulli Distribution this year as JF points out

    I vaguely remember “Bernoulli’s law of large numbers” which applied to a series of Bernoulli Trials where the observered total number of successes (a random variable X say) approximates to np as n became large where p was the probability of a successful Bernoulli Trial

    Steve R

  5. I just had a conversation with Simon the Likeable, who pointed out to me that “Bernoulli distribution” is in the Australian Curriculum. Checking back, the distribution has always been there, dating to the draft curriculum in 2012. Of course that doesn’t imply that including the distribution (= jargon) in a curriculum serves any great purpose, and it doesn’t begin to justify springing the thing on Victorian teachers and students in the middle of a plague year. But it may go some way to explaining how the VCAA clowns came up with the idea.

    1. I’d love to know *why* it was put on the Australian Curriculum. Is it there to formally ensure that a single experiment with two possible outcomes is modelled and taught. Why? To enable simple calculations of things like mean, variance …? So it’s basically a compulsory teaching aid …?

      It does have important theoretical applications and applications in logistic regression, but these applications are well-removed from any secondary school curriculum. I do not see how it has any non-trivial purpose or use in secondary schools.

      I’m genuinely interested to know if anyone can see what I can’t. I’m fully aware of examples where it can be used, but I just don’t see how that use is non-trivial at a secondary school level.

      1. Thanks, John. I can’t see any point. I *think* it’s just considered a stepping stone (more a stepping pebble) to binomial, but i’m really just guessing.

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