WitCH 86: Seeing Red (and Blue)

This one is from the 2022 Mathematical Methods Exam 1 (not yet online). We’ve decided that the question below is sufficiently bad to have earned its own WitCH. Previous comments on the question can be found on the Exam 1 discussion post.

70 Replies to “WitCH 86: Seeing Red (and Blue)”

  1. There is a lot of useless information in the question.

    I will leave it to others to determine the depth of crap.

    I have some minor gripes and some less minor gripes with the question.

    1. Yes, I agree. It is fair enough to have different types of distributions in the same question. The above question is not fair.

  2. Part a was fine, mostly a test of understanding rather simple probability concepts. b and c had potential to trip you up – I went about them by solving for a new binomial system with only 3 trials, given replacement effectively means it doesn’t matter what the first card was.

  3. The context of drawing cards is confusing, as usually that would be an example of something that’s not indepedent. I’m believe VCAA deliberately used that context solely to confuse students, in an attempt to create ”differentiation” between bands of students.

    Other than the deliberately (or at least I think so) confusing context, the rest is standard formula substitution.

    As a relevant aside, I feel sorry for EAL and international students taking VCE mathematics. They are disadvantaged inherently in these exams (and in SACs as well), as a large part of the exam is parsing through a meaningless paragraph to find the relevant info.

    1. Blue and red cards, balls in a bag, heads at an ice hockey game etc. Sampled with replacement. Is this really so confusing? I honestly don’t see the issue here.

      (OK, heads at an ice hockey game might be confusing).

      Yes, I too sympathise with ESL students. VCAA claims that the maths exams get vetted by an ESL vetter. I have my doubts. The evidence to the contrary is very strong.

      1. “The card is replaced in the deck” in normal English means taking a new card, and substituting it for the old one (which doesn’t make sense because the card is out of the deck), not sampling with replacement. Or alternatively, placing the card back into the deck—which doesn’t necessarily mean placing it randomly in the deck. Either way, it’s a pretty garbage way to write it, which they could have just avoided by saying that the card was placed back into the deck and the deck was shuffled.

        If they really wanted to connect the concept of sampling with replacement, they should have said that the deck was sampled with replacement. I didn’t even realise that was what they were implying until you mentioned it.

        1. Great comment, Ashley.

          Re: They should have said “… that the card was placed back into the deck and the deck was shuffled.”

          Good call. I hadn’t thought of that. I just assumed it was randomly placed back in the deck. I should have started wondering how the blue and red headed guys at the hockey game were being selected and replaced … (This is the problem with being too familiar with VCAA questions – you start unconsciously joining the dots to see the intended elephant, rather than VCAA clearly drawing the elephant. And sometimes their elephant is everyone else’s giraffe).

          This raises two interesting questions (and probably some crap I was previously blind to):

          How is the card drawn from the deck, and how is it replaced in the deck? These are non-trivial questions (as any polling company will tell you!!) If the top card of the deck is always drawn and when a card is replaced, it is always placed on top of the deck (such a process is not excluded by the question), then if the first card drawn is blue, then every card drawn and replaced in the deck will also be blue.

          So answers of zero and zero for parts (b) and (c) are reasonable. The question can be considered a conditional probability question after all!

          OK, I take it all back. The question \displaystyle is badly worded. VCAA should have stuck to balls in a bag.

    2. Thanks, Ashley. It’s not just EAL students.

      It is fine to have a mathematical problem with a decent amount of set-up. It is not fine to use a hundred words for a trivial set-up, much of which is then irrelevant for the majority of the question. It is crazy, and it is nasty.

      1. OK, here’s the way I see it – using cards instead of balls was a mistake – it has led to an over-complication in asking what VCAA were tying to ask in this question. Nevertheless, I \displaystyle still would have no significant issue with “Given that the first ball drawn is blue, find the …” If I’d been vetting the paper and VCAA was adamant that this is what it wanted to test, I would have strongly recommended they stop playing cards and get some balls.

        But we’re stuck with a deck of cards and the verbosity of the set-up. Is it really \displaystyle that hard to strip the question down to a binomial distribution in part (b) and (c)?

        My only gripe with parts (b) and (c) is that they ask the same thing. Part (c) should have asked something different (not just the deck to bias the probabilities).

        1. OK, so we agree that the set up for the mathematics was screwed, so that the student was made to do unnecessary and distracting and potentially confusing work. Do you also agree that:

          (1) The question was deliberately composed in a manner offering for students to trick themselves?

          (2) The actual mathematics in the question was repetitive and trivial?

          And still, you have no major problem with this question?

          1. I don’t think I’ve said that.

            I agree that the wording could be better (and using balls instead of cards would have been much better). But I don’t think the wording is bad to the point where understanding what to do is tremendously difficult.

            (1) I’ve already answered that in previous posts. There was clearly a lure, but I don’t think the lure is unreasonable. I certainly wouldn’t use the word “trick”, which implies a level of maliciousness.

            (2) I have already stated the opinion that (b) and (c) are testing the same thing and that (c) should have asked something different (maybe a switch to sampling without replacement).

            And yes, I do agree that the mathematics is trivial, I’d expect a Yr 10 student to be able to answer this question.

            My issues are:
            (a) The unsimplified fraction (I agree with RF).
            (b) How the card is drawn and how it is replaced should be stated rather than assumed. Using words such as “randomly” or “shuffled” would not have been amiss.
            (c) I’d have preferred balls in a bag to a deck of cards.

            I don’t think the question is nearly as bad, tricky, deceptive, written by drunk monkeys etc. as is being argued. Yes, it could be better written. No, it’s not unfairly written. A dark day to be sure when I’m swimming against the VCAA bashing tide (I’ve booked myself in for a full medical check-up later today).

            1. Ugh! The question is nothing but “lure”, further obscured by absurdly clumsy language. It is an appalling, pointless question.

  4. OK, I’ll begin with a minor gripe.

    VCAA and MAV (through their “meet the examiners”) have told teachers for years about the need to “tidy up” by simplifying fractions, but here in the question is a fraction, not simplified.

    OK, it makes it a tad easier to spot the 1-4-6-4-1 pattern to the numerators, but it is poor form if examiners then penalise a student for answering 4/16 rather than 1/4. Of course, we will never know if such a penalty is applied.

    I have a major gripe as well, but will save it for now.

    (JF – I don’t have an “issue” with the question per se but do think there are problems)

    1. RF, the unsimplified fraction is a fair gripe! (I’ll love to see your major one, unless I’ve already mentioned it in my second response to Ashley). And the crazy thing is that there was absolutely no need not to simplify. The only thing consistent about VCAA is its lack of consistency.

        1. If it was just this paper, I would say the fraction is fine.

          But given the history of being told (by examiners) that students need to simplify their fractions…

          Hence a minor gripe.

          1. If it were just this paper …

            The unsimplified fraction is interesting. I can only think the examiners were trying to be helpful, to give the probabilities exactly as binomial would spit them out. But it’s pretty infantilising, given part (a) is trivial. Also, given (a) is not then used, it seems pointlessly over-helpful.

    2. Marty has said that VCAA are aware of his blog. So they should be following this post. So VCAA, if you \displaystyle are reading:

      There’s going to be big, big trouble if students get penalised for not simplifying fractions in this question. And there’s going to big trouble if the Examination Report doesn’t make some mention of this and specifically state that not simplifying in this question is a one-off exception.

      1. “following” is way too strong. By memory, I wrote “aware of”.

        In any case, there’s no way VCAA will penalise unsimplified fractions in (a).

        1. If they are aware, they would be crazy not to be following the discussion on the exams. Oh … hmm hm .. yep, OK.

          1. I think MacNeill is clearly paying more attention to teacher-student-gadfly opinion than his predecessors. But I also wouldn’t over-hope it.

    1. Yes. It is a huge issue, and an issue schools should be addressing rather than worsening. But you don’t address it by giving students an exam question set-up worded in the longest and stupidest possible manner.

  5. OK, major gripe.

    “the process is repeated four times.” OK, fair enough.

    “the next three cards drawn…” is that AFTER four cards are drawn or does this complete the four drawn cards…?

    The grammar in this question IS INSANELY BAD (and this is coming from someone who is normally oblivious to poor grammar).

    1. Sorry, RF. We agree on many things but I disagree with you on this one. You’re told the first card is drawn (and replaced) and then asked what happens with the next three cards drawn (and replaced). I honestly don’t see the issue with the wording here.

      1. When working it through as a question, I assumed it was just four cards drawn.

        It was my second and third reading that raised my hackles, so to speak.

        Not an error, but far from well written.

    2. I don’t think the sentence is ambiguous. Note the word used is “performed”, not “repeated”.

      But I think it is a bad sentence, badly placed. If nothing else, the “this” is doing too much work.

  6. “The color of any card drawn is indep of the color of any other drawn card”…..
    2nd place in the comp for most clumsy phrase?
    I feel I hace to dissect it to be sure it is even correct.
    “Given the 1st card drawn…” …real question or a riddle? O.K. Testing if students apply conditional automatically or go for a 3-trial binom distr.
    But then, WHY TWICE in one question???

    Why is it in given scenario the first? What about 2nd? 3rd? being of known color?

  7. I’m confused with Pr(X=x) being greater than zero (ie. 1/16) when x = 0. It doesn’t make sense to me. But, then again, probability is not my strength…

    1. X = 0 means getting zero blue cards. There’s definitely a positive probably of that (i.e. of getting all red cards.)

  8. Student who raised this wording on the original post here – these seem to be the only marks I lost (on two papers appallingly devoid of differentiating marks)

    The question stem does say 4 cards are drawn, however in previous NHT and regular VCAA papers the distribution being investigated has changed in sub questions, effectively ‘contradicting’ the stem (if the stem is interpreted as applying across the whole question). As such I hesitated to do what seems to be the correct method of redefining the binomial with 3 trials and then simply finding Pr(X=1), instead interpreting it as the first card out of 3 drawn being blue (which also involves redefining this but then reducing the sample space further).

    Probably not an actual error but worded in a predictably ambiguous way especially since the logical conclusion that the question stem applies for the entire question has been contradicted by VCAA in the past

    1. Thanks for your comment, Student.

      There is no doubt that over the last 20 years VCAA has rightly gained an inglorious reputation for pedantry, inconsistency, mistakes and (im)plausible deniability, which in turn has led to mistrust, doubt and uncertainty in the minds of students and teachers. However ….

      As I’ve said in another post, you have to play the question, not the man. You cannot second guess motives, see phantoms where there are none etc.

      I think a major part of the issues people have had with this question (not withstanding my own issue commented on earlier) stems from a (justified) bias against VCAA, second guessing what its questions mean and not quite believing what the question says. That’s not the fault of the question, but it does show what happens when there is (justified) mistrust.

        1. So what does a student do? Have the guts to say they think the question is wrong and then hope like hell that it is? It would be be an extraordinarily brave student that said

          “I think the question is wrong. Here is my answer based on my interpretation of the question: …”

          (However, I am aware of this advice being given in certain types of advanced exams – Membership exams in certain professions like Medicine, Veterinary Science etc).

          So I have to disagree, Marty. You’ve got to play the question, answer it as best you can. Whether the question is right or wrong. When was the last time VCAA conceded a question was wrong (and I don’t mean a multiple choice question, I mean a question like this Question 4)?

          1. Of course a student in an exam has no choice but to interpret and answer the question as best they can. But you also wrote “you cannot second guess motives”, and that is plain wrong. Given VCAA writes so poorly, and often enough incorrectly, a student must second guess motives.

            1. A student second guessing motives (*) in an exam is a recipe for disaster. I think we saw evidence for this with comments about Exam 1 (“the question was too easy, I was looking for the trick and over thought it”)

              Students should follow the immortal advice of countless Sergeant Major’s: “Shut up and soldier, soldier!!!”

              * Unless this means trying to assume what VCAA intends. And I agree that having to assume the intention of a question in order to answer it is diabolical and should never happen.

                1. I always tell students to include the “+C” regardless of whether they’re told to “Find an …” or “Find the …” It’s never wrong.

                  I have another reason for this – I don’t like them finding \displaystyle an antiderivative of something like sin(x) cos(x) and then getting confused by the differences in equivalent answers …

                  It’s only when they’re finding an antiderivative in order to calculate a definite integral that I say forget about the “+C”.

                  1. Sure. And what if you were asked to find “the rule” for “an antiderivative”? Just speaking hypothetically, of course.

                    1. Of course it’s only hypothetical.

                      I always tell students to include the “+C” regardless of whether they’re told to “Find an …” or “Find the …” It’s never wrong. In this case, it’s “an”.

                      I have a theory on the “the rule” crap. I’ll guess VCAA did this to stop the small minority of students from also stating the domain of the antiderivative. Trying to help but just making things more hamfisted.

                  2. Forgive me for lengthening this aside…

                    Maybe I’m missing why adding a +C would do anything to fix the whole antiderivative mess, but “finding an” antiderivative as

                    F(x)

                    is no worse than

                    F(x) + C

                    where C is some constant. Because you aren’t finding the constant, and in fact it doesn’t matter what constant you put there, they will all differentiate to the same thing. So why include it at all?

                    Is the point I’m missing that when students are asked to do this the question wants them to give a general form for any antiderivative, and is asking literally for the wrong thing? So to get the marks, they need to answer a question that wasn’t asked? That’s messed up if so.

                    1. \displaystyle The antiderivative of 2x is \displaystyle x^2 + C where C s an arbitrary constant. It’s a family of functions.
                      \displaystyle An antiderivative of 2x is \displaystyle x^2 + C where C has been given any specific value you want, including zero (in which case you have \displaystyle x^2) (or you can just keep it as C). It’s any member of the family.

      1. I don’t think it’s deliberately misleading. I think it’s deliberately testing whether a student understands independence. I think it will turn out to be a very polarising question.

        1. It is deliberately offering for a student to mislead themselves, and contains little else other than poor wording.

  9. Let’s say one wanted to ask this question in a better way. Would the following have been adequate?

    “Consider a deck of cards containing the same number of red cards and blue cards, and no other colours. Four cards are randomly drawn with replacement.”

    Part a could have been: “Complete the following table showing the probabilities of drawing 0, 1, 2, 3, or 4 blue cards.” (I don’t think there’s any need to introduce random variable notation here).

    Part b could then have been: “The first card drawn is blue. Find the probability that two of the next three cards drawn are red”.

    Part c could then have been: “Now suppose there are twice as many red cards as blue cards in the deck. Again, four cards are randomly drawn with replacement. The first card drawn is blue. Find the probability that two of the next three cards drawn are red”.

    1. Thanks, SRK. I’d have to think if I wanted to nitpick and try to improve what you’ve written. But your suggested wording is very short and very clear, and is obviously way, way, way better than VCAA’s word swamp.

      1. Yep that wording would’ve been great if I had read that in the exam. If they wanted to test understanding of independence between Bernoulli trials then that’s great, but then actually test that rather than the ability decipher if that is what is being asked or something else. Even if they wanted to keep the weird “given” wording so as to not make it obvious that only the last 3 trials (out of 4) are to be considered, they could word it something like “Given that a card is drawn, seen to be blue and then returned to the deck, find the probability that 2 out of the remaining 3 cards are red”. Not perfect but the “given” pretext can easily be interpreted as pertaining to whatever the main clause of the sentence says rather than interpreted as a seperate event. So a clearer sequence of events and the word “remaining” would have helped

    2. This would fix the wording issue but the math remains definitely weird. I’d like it better if (a) was involved in (b), and I’d like it even better if one of (b) or (c) had actual conditional probability. I don’t mind testing for independence, but I think this would give a more coherent (and slightly more difficult) question overall.

      1. Conditional probability would only be non-trivial if the sampling was without replacement. I have said from the get-go that (c) should test something different to (b) and that sampling without replacement was an obvious choice. Including conditional probability within that is a good idea. But then, “Given that the first card drawn is blue” really does become a trap! And I would certainly agree that it would make the use of this phrase an unreasonable lure in part (b).

  10. It seems that only Student and Marty have mentioned the real problem here. If I was sitting this exam I would have answered the three parts with 3 different cases of the binomial probability formula: (a) n=4, p=1/2, (b) n=3, p=1/2, (c) n=3, p=2/3. But then I would be nervous for the following reasons.
    (i) It’s strange that the same material is examined 3 times.
    (ii) For questions worded like this, one expects the answer to the first part to be somehow used later.
    (iii) Usually a probability question starting with “Given that” involves the Bayes formula or equivalent.
    Then I would start to obsess that I must have misread the question and was really required to perform some Bayesian analysis involving part (a) to answer part (b) and/or (c). Maybe that is the case?

    PS. Yes the word “replaced” should be replaced! by “put back”, but plain English sounds amateurish to beginners. And any damage is rectified later where the constant probability of 1/2 is specified, and the independence of events asserted.

    1. Tom, you raise reasonable points. BUT, and this is where I will sound nasty, you have overthought the question looking for phantoms. Yes, VCAA have encouraged this paranoia. Nevertheless, the question was there to be answered, not to be the subject of over-thinking, second guessing, looking for the trap etc. As the Sergeant Major says ….

      1. Point taken. I was reporting exactly how I felt while doing this question. I suspect that some of the better students would have been puzzled too?

        1. Tom, you’re entitled to your feelings. And you’re not alone in having those feelings. The fact is that you shouldn’t have to have those feelings. No-one should. And you wouldn’t have those feelings if VCAA had done its job properly in the past. VCAA’s reputation (*) is such that no matter how easy or obvious something is (and I’m not saying this question was obvious), teachers and students will be mistrustful and look for the catch, the trick. VCAA has created a culture causing teachers and students to doubt and second guess.

          * I think the new Maths Manager is trying hard to change this reputation and restore trust. He has to undo the last 20 years – no mean feat.

  11. One amusing thing for me is that the thing that confused me when first reading this question (which I mentioned earlier on the other post, but didn’t elaborate) hasn’t been mentioned at all yet. I think this particular part is just horribly difficult to parse nonsense.

    The wording throughout the question is bad, but I’ve complained a lot about wording in exams and have been told it is standard. For us I guess it isn’t hard to see what they want you to do. But that doesn’t mean it is not bad.

    As to the actual mathematics in the question, well, there isn’t much there. Something that I didn’t appreciate earlier is how much the math that is in the question adds to its overall oddness. It’s genuinely weird, like layers of misdirection and confusion, with a simple task at the end of the rainbow. Just a really bizarre question that I’m sure left students feeling uneasy and off-balance. For that, I hate it.

  12. Came across this question today for the first time from a student.

    My initial response to question (b) was to interpret is as

    Pr (X = 2 | X>=1)

    “Given that the first card drawn is blue (meaning the value of X will be at least equal to 1), what is the probability that two of the next three cards are red (meaning that X will end up equalling 2 – the first card is blue, and one of the next three cards is blue)”

    When calculating Pr (X = 2 | X>=1) the answer is 6/15 or 2/5.

    I accept the answer is 3/8 (basically ignoring the apparent conditional element of the question, and looking at it as a three stage tree diagram) but wondering if anyone can draw some light on the error in my logic as outlined above? Been pondering this one all day.

    Cheers

    1. Hi, Mark.

      The “condition” is not that X ≥ 1, i.e there is at least one blue card, but that the first card is blue. So, your conditional argument properly reduces to the tree you also did.

    2. Since the trials are independent, what happens in the first trial doesn’t effect what happens in the next three trials. And it’s only the next three trials that we’re interested in. Three cards are drawn and replaced each time.

      Despite what the wording might suggest, there is no conditional probability here. What happens with the first card is a *ahem* blue herring.

      So it’s binomial with n = 3 and p = 1/2. You want the probability that exactly two of the three cards are red.

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