WitCH 46: Paddling in the Gene Pool

The question below is from the first Methods exam (not online), held a few days ago, and which we’ll write upon more generally very soon. The question was brought to our attention by frequent commenter Red Five, and we’ve been pondering it for a couple days; we’re not sure whether it’s sufficient for a WitCH, or is a PoSWW, or is just a little silly. But, whatever it is, it’s pretty annoying, so what the hell.

30 Replies to “WitCH 46: Paddling in the Gene Pool”

  1. Hi Marty,
    Being a student that undertook the very same exam two days ago, I’ll have a stab at it.

    Part A) seems like a reasonable computation. However, part B) demands a very specific form for probability which seems very cluttered. (Would you prefer something like 9/25 or 3^2 / 5^2 ?). It was only pure algebraic preference that I did manage to attain this specific form in the first try, but I knew many other students who fell victim and evaluated the indices before actually computing the probability without a clue on how to condense it. (e.g turning 3^2 in the numerator into 9 which ultimately won’t give you the pedantic form required). It was only pure luck (and a bit of algebraic recognition!) that I did not follow such paths and found the form required.

    Got the marks, but only out of luck. There shouldn’t be a reason why you can’t evaluate the numbers out.

    1. Hi, Tsui. Good to hear from you and congratulations on surviving Methods. The ridiculousness of part (b) is confirmed by the story of your fellow students, who (presumably) worded out the probability in a perfectly acceptable manner, and then had to work to get their (presumably) fine answer into the arbitrary form demanded by the examiners. All for 2 marks. This is complete madness.

  2. I’ll add here that what annoyed me was entirely my own fault. I worked out the fraction and then read that they wanted a, b, c values.

    I just felt this was a bit… odd (except, upon reflection it is entirely consistent with what VCAA has done in the past).

    1. RF, your annoyance may have been very partially your own fault, but it definitely wasn’t entirely your own fault. Yes, one should first read the entire question carefully and, yes, VCAA has a well-known fetish for students answering “in the form of”. But the “form” in this question is needless, arbitrary and absurd.

  3. On a side note, I wonder when VCAA will ask these questions better, or even stop asking for such specific forms that could have multiple representations. Another set of values (a,b and c) that also give 73/203 (the actual value) would be a = 96, b = 40 and c =16. At least they didn’t ask for the values of a, b and c, otherwise they would have really screwed themselves over. I don’t see why students would have to answer in such a specific way, aside from simulating artificial difficulty (or whatever you would like to call it…) While I’m at it, has something similar happened on exam 2 where one could use a CAS calculator to determine a separate set of values when asked to express something in another form?

    1. Thanks, Sai, and I agree entirely. Mathematicians appreciate the form of an expression in certain contexts, and can obsess over it, since the form can be critical to gaining insight into the underlying problem. But such a form is usually unique or clearly and tightly constrained, and fussing about such a form requires a clear purpose. The VCAA seemingly never concerns itself with uniqueness, which is gauche, and the form demanded in (b) is way worse than purposeless.

  4. I don’t mind this question. In part (b), if you don’t multiply the factors (my preference), the required pattern falls out. So the wording actually makes it a little easier to my mind.

    I was more concerned with the preamble; the idea that the probability of someone having a gene being independent of say, their mother having it, seemed far-fetched. Then I realised that you could select a population with no close relatives. Or use a population with identifiers removed so our estimates of the probability cannot use relationships. Does the wording allow this?

    1. Re: Context. I gave up long ago expecting the VCAA mandatory fabricated context to have any bearing to reality. It’s like a movie, you just have to suspend your sense of disbelief and go along with it. (Except the movies are generally done much better and are way more fun). So, apart from a sigh, I don’t have a strong objection to the genetics context.

      Re: The form of solution for part (b). This was probably done so that students could avoid ‘excessive’ arithmetic calculation in coming up with 73/203. So I don’t see it simulating artificial difficulty, I see it attempting to make life a bit easier. Sai is correct in his stated set of alternative values. Indeed, there are an infinite number of possible sets of values for (a, b, c) and all would/should be be acceptable. But the point is that none except the (6, 5, 2) set emerge ‘naturally’ from the obvious calculation.
      @RF: Sorry old chum, but the exam in my view was too long, so unfortunately I doubt any student had a chance to channel their inner smart-arse. And unfortunately I think VCAA impacts the epigenetic process in such a way that being a smart-arse does not get expressed …

      So maybe I’m just too jaded to see anything too bad with this question. But there were a couple of things in other questions that irked me – not errors (technical or otherwise), just wording that I think could encourage errors. And, as already said, I think the exam was (marginally) too long.

      So in the words of a great man: “All in all not bad by Methods exam standards, but there’s a few things to slap”.

      PS: Good luck, Tsui M. Sounds like you might have done well in this exam.

      1. Totally agreed JF – I can dream though…

        By comparison, I found the SM paper 1 a lot more… “as expected” than the MM paper 1. Still a few things to not love about it (the vector resolute question in particular).

        1. Not wanting to get off-topic but yes, I agree. It was *mostly harmless* (but, again, long). I didn’t mind the vector resolute question but I thought recognising the factorisation in part (a) was a bit stiff (those who used the quadratic formula had ‘messy’ arithmetic computations) and the explicit rejection of the extraneous solution might elude many students (although the wording of part (b) gives a strong hint that there’s only one solution).

          1. Agreed about the factoring (more or less forced), but the question states that m is an integer, so I’d say the only reason rejecting the extraneous solution would elude students would be carelessness or time-pressure.

          2. Actually rejecting the extraneous solution is not a problem because the preamble says m is an integer. (I guess VCAA figured recognising and rejecting an extraneous was too difficult).

            Edit: Aha I see SRK pre-empted my blind spot above.

    2. Thanks, Tom. The fact that the wording in (b) made it easier to your mind doesn’t negate the fact that it much it much harder for other reasonable minds.

      As for the preamble, it is absolutely insane and it is the reason I decided this question must have its own post. Genes and inheritance is the archetypal example of characteristics being dependent. The idea of using such a scenario for a question on independence is hilariously stupid, and the hilarity and stupidity is only lessened epsilon by the fact that one can imagine a “certain population” where no one is related to anyone else.

  5. I don’t like the wording “The probability of a person having this gene is independent …”. In probability theory, we talk of events being independent, not probabilities.

    I did not like the wording “given that at least one of those people…”. I would have written “given that at least one of the four people…”.

    It strikes me that asking to students to express the conditional probability in the given form is a pointless exercise. Probability and statistics are in the course because it is generally agreed that they are useful mathematical methods. (Ask Dr Boaler!) I see no point in asking students to express the answer in this way. When setting a question for a test, one should always ask, “What is the point of the question?”

    1. Since the exam is calculator-free, the point (as I see it) “in asking students to express the answer in this way” is to spare them ‘messy’ arithmetic computation. One could argue that the question shouldn’t require ‘messy’ arithmetic computations – nevertheless, if the intent is to test (among other things) whether a student can calculate non-trivial probabilities from a binomial distribution and use them in a conditional probability context, then it’s probably unavoidable. Maybe students should be given a choice: “Express your answer either in the form … or as a simplified fraction.”

      btw I use the word ‘messy’ in the context of a significant minority of the cohort struggling to correctly calculate even 23×23 without a calculator. In fact, I bet the examination report (when we get to see it some time in Term 3) reveals a significant number of students not able to correctly calculate simple combinatorials. (Instead of the Bernoulli Distribution being uselessly included on the formula sheet, the first few rows of Pascal’s Triangle should have been included). But now that universities are allowing entry into engineering without Maths Methods, these issues might disappear (along with Maths Methods) …

      1. Another approach is to use p instead of 3/5 and leave n=4. Let X be the number of people in the sample with the gene.

        (i) Prove that P(X \geq 1) = 1 - (1-p)^4\,.

        (ii) Prove that P(X=2|X \geq 1) = \displaystyle{\frac{6p^2(1-p)^2}{1-(1-p)^4}}\,.

        This exercise assesses a student’s knowledge of the binomial distribution and conditional probability (content) and reasoning ability (proficiency). Since proficiencies are central to the curriculum, they should be assessed as is the content.

        1. Thanks, Terry. Yes, the question would have been much better if it had been composed in some algebraic form. This sort of screw-up is standard.

      2. John, the arithmetic is trivial and the demanded form of the answer is ridiculous. If the VCAA were concerned about the trivial arithmetic, then they should have jiggled with the numbers to make the trivial arithmetic even more trivial.

    2. Well spotted Terry.

      More evidence that the exam setters (or proof-readers, or both) don’t get the subtleties of probability (although they seem to be worse in statistics!)

      1. Indeed, TM.

        @RF: The exam-setting panel consists of teachers, not experts. And the vettors are probably teachers as well. In my experience, probability and statistics is poorly understood (and hence poorly taught) by many teachers (which is why so many students struggle with it and just don’t like it). It’s an area of VCE mathematics more than any other that requires more than just a superficial understanding from reading the school textbook.

        In my opinion, teachers of VCE Methods and Specialist should be required to either have studied probability and statistics to at least 1st year uni level, or attend extensive professional development on the subject and get a non-trivial ‘micro-credential’. Their understanding should not be based on reading the school textbook and memorising routine examples (which I’m certain is mostly the case with the statistics added to Specialist Maths).

        1. John, I think the problem with stats is that teachers can’t wing it in the manner they (sort of) can with other VCE maths topics.

          There are *plenty* of teachers who took too few maths subjects, or took only shithouse maths subjects, or scraped through their maths subjects, or all three. But, if the topic is reasonably solid and coherent, a diligent and intelligent teacher can learn on the job and get by; it’s not optimal and they’ll be tricked on occasion, but it works. My “unqualified teacher” mother was a fine example of this. VCE stats, however, is neither solid nor coherent, making it pretty much impossible to learn on the job.

          The solution is not to train teachers in stats: the solution is to get rid of a fucking useless topic.

          1. I totally agree. But which solution is more likely to be possible? (Maybe yours IF there’s a change of leadership in VCAA).

          2. Marti,

            It is not the statistics or probability branches of mathematics that are useless …just the way these subjects are trivialised in Methods short answer questions.

            The question above is answered neatly by TM and the rest is unnecessary algebraic manipulation

            Steve R

  6. Hi Terry
    I see you are up late – or very early. As an amateur in probability (studied physics instead) I bow to your opinions.
    Wording exam questions is hard, especially in the messy world of statistics. This site would do a great service if we could agree on some examples of good questioning.

    1. Nonsense. That’d be just as valuable as giving ScoMoFo and his henchman some examples of good governance. The VCAA doesn’t give a flying fuck about “good questioning”.

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