WitCH 33: Below Average

We’re not actively looking for WitCHes right now, since we have a huge backlog to update. This one, however, came up in another context and, after chatting about it with commenter Red Five, there seemed no choice.

The following 1-mark multiple choice question appeared in 2019 Exam 2 (CAS) of VCE’s Mathematical Methods.

The problem was to determine Pr(X > 0), the possible answers being

A. 2/3      B. 3/4      C. 4/5      D. 7/9      E. 5/6

Have fun.

13 Replies to “WitCH 33: Below Average”

  1. Defining f(x) to be equal to p(x) between -a and b and 0 elsewhere seems redundant. Why not simply say that p is the probability density function?

    Further, p(x) is never defined other than via the graph, so we don’t technically know that it’s pieced together from two linear functions. The test should state/mathematically-imply in the definition of p that p is pieced together from two linear functions.

  2. It’s a curious question, because the only probabilistic part is noting that the area under the given graph must be 1, hence the average value given shows that a+b = 4/3. Then it’s just geometry and algebra. The area, as a triangle on the left and a trapezoid on the right, is a^2 + (b/2)(2a+b) = a^2 + ab + b^2/2. Since this is equal to one, we multiply by 2 to obtain 2a^2 + 2ab + b^2 = 2, and note that the right hand side can be rewritten to obtain a^2 + (a + b)^2 = 2. Since a+b = 4/3, the answer is 1 – a^2 = 7/9.

    I have several points to make:

    1, This seems like quite a lot of work for 1 point.

    2, I have no idea of the pedagogy behind this. What understanding are the examiners intending to test, and how does this question actually test this understanding? (Mind you, this query could be made of many VCE maths exam questions.)

    3, My solution above has b being independent of a, whereas in the picture it seems that b < 2a. If you start by assuming this, and obtain the same answer, are you wrong? But then, as a multiple choice question there is no way of checking working.

    4, Multiple choice questions, unless very carefully designed, have no place in mathematics assessment. They reinforce the idea that mathematics is about “answers” and “getting it right”, instead of demonstrating reasoning and argument.

    5, It’s a dog’s breakfast of a question: a mish-mash of ideas, and really really poorly examined. A perfect example of a question which should not be multiple choice.

    You could probably go deeper into the setup, as edderiofer has done above, and pull apart some of the sloppy definitions. But as we all know, the only way to make sense of VCAA mathematics is to take a general level of sloppiness on board and so give everybody a large measure of doubt.

    1. Thanks, amca01, and I agree entirely. I would just shorten your point 5: A perfect example of a question which should not be.

  3. @edderiofer @amca01 – Agreed; Further,wrt sloppiness, the diagram is a sketch not a graph, in the conventional sense that a graph is a representation of the function in Cartesian coordinates with respect to linearly-scaled orthogonal axes. If correctly drawn, the lengths of the abscissa and ordinate of (b,b) would be in the same ratio as the length of the abscissa of (-a,0) and 0.5*length of ordinate of (0,2a). ie Point (b,b) should lie on a line through the origin with half the slope of the left-hand side arm of the graph.

  4. Alternatively, an astute student could realize that as the answer is independant of b, they can choose any value they like, say b=1. Then a=1/3 from which the answer is immediate. And all my points still stand, with the extra possibility of confirming the perception that mathematics is really all about learning a mass of nifty tricks.

    1. I think I see what you’re getting at here… and will now go and try a few values to see. I suspect that if it were not for the left hand part of the graph, this might work since b acts as a scale factor.

      But it does raise a valid point about areas under a graph more generally, since Methods examiners have been known (at those “meet the assessors” sessions) to suggest that “guessing a function” is sometimes valid on MCQs.

  5. By the way, talking about VCAA Exam questions ….. At the risk of being off-topic, I’m sure everyone has heard the wonderful news:

    VCAA have at long last made the marking scheme for its exams public. Finally. Unfortunately, it’s not freely available. You have to pay the MAV a sum of money to get access. Apparently the VCAA Assessors are no longer bound by confidentiality and can disclose the marking scheme through the MAV VCAA Exam Solutions. A cosy arrangement where the Assessors get to make extra money, MAV make money and teachers get the VCAA marking scheme. Everyone’s happy.

  6. Hi, JF, and everyone. I’ll post on the MAV-VCAA solutions thing shortly. Please leave thoughts and comments for that post.

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