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. Continue reading “WitCH 86: Seeing Red (and Blue)”
Tag: probability
WitCH 85: The Continuation of MAV’s Trials
With Methods exams next week, this one’s kinda important.
We try to avoid critiquing, or even being in the same room as, third party VCE practice exams. They are invariably clunky and weird, with plenty to criticise, but they matter infinitely less than the yearly screw-ups of the official exams.
Even MAV trial exams we do our best to ignore. Yes, the MAV is (too) closely aligned with the VCAA (with a number of people in conflicted, dual roles), and so MAV has a significantly greater professional and moral obligation to maintain high standards. But still, third party is third party, and we try our best to just ignore MAV’s nonsense. On occasion, however, MAV’s nonsense matters sufficiently, or is simply sufficiently annoying, to warrant a whack.
Continue reading “WitCH 85: The Continuation of MAV’s Trials”
GitS 2 : John Friend – A Lack of Confidence
This post, on confidence intervals, is by frequent commenter, John Friend. It is the second of our guest posts; the first, by Anthony Harradine, is here. A version of John’s post is also available as a PDF, here. Continue reading “GitS 2 : John Friend – A Lack of Confidence”
WitCH 73: Independent Thinking
Here is an old Methods one for a change, from the 2011 Exam 2. The examination report gives the answer as E and indicates that 15% of students selected this answer.
PoSWW 20: Unconventional Wisdom
This one comes courtesy of frequent commenter, John Friend. It is an example from Cambridge’s Mathematical Methods 34.
UPDATE (19/08/21)
It amazes me at times what does and does not concern some commenters. That’s not intended as a criticism. Well, it is, but it isn’t. And, it is. It’s complicated.
Cicchetti’s Random Shit
Readers will be aware that Trump and his MAGA goons have been pretending that Joe Biden stole the US election. They’ve been counting on the corruptness of sufficient judges and election officials for their fantasy grievances to gain traction. So far, however, and this was no gimme, the authorities have, in the main, been unwilling to deny reality.
The latest denial of the denial of reality came yesterday, with the Supreme Court telling Texas’s scumbag attorney general, and 17 other scumbag attorneys general, and 126 scumbag congressmen, to go fuck themselves. AG Paxton’s lawsuit, arguing to invalidate the election results in four states, was garbage in every conceivable way, and in a few inconceivable ways. One of those inconceivable ways was mathematical, which is why we are here.
As David Post wrote about here and then here, Paxton’s original motion claimed powerful statistical evidence, giving “substantial reason to doubt the voting results in the Defendant States” (paragraphs 9 – 12). In particular, Paxton claimed that Trump’s early lead in the voting was statistically insurmountable (par 10):
“The probability of former Vice President Biden winning the popular vote in the four Defendant States—Georgia, Michigan, Pennsylvania, and Wisconsin—independently given President Trump’s early lead in those States as of 3 a.m. on November 4, 2020, is less than one in a quadrillion, or 1 in 1,000,000,000,000,000.”
Similarly, Paxton looked to Trump’s defeat of Clinton in 2016 to argue the unlikelihood of Biden’s win in these states (par 11):
“The same less than one in a quadrillion statistical improbability … exists when Mr. Biden’s performance in each of those Defendant States is compared to former Secretary of State Hilary Clinton’s performance in the 2016 general election and President Trump’s performance in the 2016 and 2020 general elections.”
On the face of it, these claims are, well, insane. So, what evidence did Paxton produce? It appeared in Paxton’s subsequent motion for expedited consideration, in the form of a Declaration to the Court by “Charles J. Cicchetti, PhD” (pages 20-29). Cicchetti’s Declaration has to be read to be believed.
Cicchetti‘s PhD is in economics, and he is a managing director of a corporate consulting group called Berkeley Research Group. BRG appears to have no role in Paxton’s suit, and Cicchetti doesn’t say how he got involved; he simply writes that he was “asked to analyze some of the validity and credibility of the 2020 presidential election in key battleground states”. Presumably, Paxton was just after the best.
It is excruciating to read Cicchetti’s entire Declaration, but there is also no need. Amongst all the Z-scores and whatnot, Cicchetti’s argument is trivial. Here is the essence of Cicchetti’s support for Paxton’s statements above.
In regard to Trump’s early lead, Cicchetti discusses Georgia, comparing the early vote and late vote distributions (par 15):
“I use a Z-score to test if the votes from the two samples are statistically similar … There is a one in many more than quadrillions of chances that these two tabulation periods are randomly drawn from the same population.
Similarly, in regard to Biden outperforming Clinton in the four states, Cicchetti writes
“I tested the hypothesis that the performance of the two Democrat candidates were statistically similar by comparing Clinton to Biden … [Cicchetti sprinkles some Z-score fairy dust] … I can reject the hypothesis many times more than one in a quadrillion times that the two outcomes were similar.”
And, as David Post has noted, that’s all there is. Cicchetti has demonstrated that the late Georgia votes skewed strongly to Biden, and that Biden outperformed Clinton. Both of which everybody knew was gonna happen and everybody knows did happen.
None of this, of course, supports Paxton’s claims in the slightest. So, was Cicchetti really so stupid as to think he was proving anything? No, Cicchetti may be stupid but he’s not that stupid; Cicchetti briefly addresses the fact that his argument contains no argument. In regard to the late swing in Georgia, Cicchetti writes (par 16)
“I am aware of some anecdotal statements from election night that some Democratic strongholds were yet to be tabulated … [This] could cause the later ballots to be non-randomly different … but I am not aware of any actual [supporting] data …”
Yep, it’s up to others to demonstrate that the late votes went to Biden. Which, you know they kind of did, when they counted the fucking votes. As for Biden outperforming Clinton, Cicchetti writes (par 13),
“There are many possible reasons why people vote for different candidates. However, I find the increase of Biden over Clinto is statistically incredible if the outcomes were based on similar populations of voters …”
Yep, Cicchetti finds it “incredible” that four years of that motherfucker Trump had such an effect on how people voted.
What an asshole.
WitCH 47: A Bad Inflection
The question below is from the first 2020 Specialist exam (not online). It has been discussed in the comments here, and the main issues have been noted, but we’ve decided the question is sufficiently flawed to warrant its own post.
UPDATE (10/09/21) For those who’d placed a wager, the examination report (Word-doc-VCAA-stupid) indicates that a second derivative argument was expected. Hence, thousands of VCE students no longer have any sense of what VCAA means by “hence”.
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.
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.
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
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.Update (04/07/20)
Who writes this crap? Who writes such a problem, who proofreads such a problem, and then says “Yep, that’ll work”? Because it didn’t work, and it was never going to. The examination report indicates that 27% of students gave the correct answer, a tick or two above random guessing.
We’ll outline a solution below, but first to the crap. The main awfulness is the double-function nonsense, defining the probability distribution 
in terms of pretty make the same function 
. What’s the point of that? Well, of course is defined on all of 
and is only defined on ![\boldsymbol{[-a,b]}](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-542a7cba934ae34488577b5116dbaa31_l3.svg "Rendered by QuickLaTeX.com")
. And, what’s the point of defining on all of ? There’s absolutely none. It’s completely gratuitous and, here, completely ridiculous. It is all the worse, and all the more ridiculous, since the function isn’t properly defined or labelled piecewise linear, or anything; it’s just Magritte crap.
in terms of pretty make the same function . What’s the point of that? Well, of course is defined on all of and is only defined on . And, what’s the point of defining on all of ? There’s absolutely none. It’s completely gratuitous and, here, completely ridiculous. It is all the worse, and all the more ridiculous, since the function isn’t properly defined or labelled piecewise linear, or anything; it’s just Magritte crap. To add to the Magritte crap, commenter Oliver Oliver has pointed out the hilarious Dali crap, that the Magritte graph is impossible even on its own terms. Beginning in the first quadrant, the point 
is not quite symmetrically placed to make a 
angle. And, yeah, the axes can be scaled differently, but why would one do it here? But now for the Dali: consider the second quadrant and ask yourself, how are the axes scaled there? Taking a hit of acid may assist in answering that one.
is not quite symmetrically placed to make a angle. And, yeah, the axes can be scaled differently, but why would one do it here? But now for the Dali: consider the second quadrant and ask yourself, how are the axes scaled there? Taking a hit of acid may assist in answering that one.Now, finally to the problem. As we indicated, the problem itself is fine, it’s just weird and tricky and hellishly long. And worth 1 mark.
As commenters have pointed out, the problem doesn’t have a whole lot to do with probability. That’s just a scenario to give rise to the two equations,
1) 
and
2) 
The problem is then to evaluate
*) 
or, equivalently,
**) 
The problem is tricky, not least because it feels as if there may be an easy way to avoid the full-blown simultaneous equations. This does not appear to be the case, however. Of course, the VCAA just expects the lobotomised students to push the damn buttons which, one must admit, saves the students from being tricked.
Anyway, for the non-lobotomised among us, the simplest approach seems to be that indicated below, by commenter amca01. First multiply equation (1) by 2 and rearrange, to give
3) 
Then, plugging in (2), we have
4) 
That then plugs into **), giving the answer 7/9.
Very nice. And a whole 90 seconds to complete, not counting the time lost making sense of all the crap.