# WitCH 129: Losing Your Lunch

Onward, ho! We’ll get going with the recent Year 7 NAPLAN questions, beginning with the 2017 test.* This first one, from the calculator section, also appeared on the 2017 Year 9 test.**

*) If anyone has access to the 2018 Year 7 test, or any recent Year 3 or Year 5 tests, we are interested. (Of course, do not attach any tests to comments.)

**) We had flagged it for WitCHing with the other Year 9 questions and then simply forgot about it.

## 18 Replies to “WitCH 129: Losing Your Lunch”

1. Back in Black says:

Wow!
If it’s a given fact that Lucas only buys lunch ONCE every five school days, then we better know the last time that Lucas bought lunch before we figure out the probability that he buys lunch TODAY.

(Eg. Today is Tuesday. Lucas bought lunch yesterday. So the probability of him buying lunch today is zero).

And if he hasn’t bought lunch on Monday – Thursday, he’s certain to buy it on Friday).

1. Ally says:

This probably seems like a stupid question, but this is with the assumption that it’s once every 5 school days across the same week, right?
(I know with, say, business days, weekends are deliberately excluded, hence the question)

1. marty says:

Hi, Ally, not a stupid question. That is, yours is not a stupid question: the NAPLAN question is monumentally stupid.

In some sense it doesn’t matter. You could consider it 1 out of every 5 days in a given school week, or 100 days out of 500, or whatever. The fundamental problem is that there is in an unstated, and implausible, assumption of randomness to the days.

Imagine, for example, that Ally has to go to work early every Monday, and so doesn’t have time to make the kid’s lunch that day. Then every Monday Luke buys lunch at the school. So, one out of every five schooldays, Luke buys lunch. But that doesn’t tell us the probability of buying lunch “today”: it’s 1 if it’s a Monday and 0 if it’s another day, and it’s 1/5 if we’re drunk and don’t know what day it is.

1. Cathy says:

Is the assessor a Bayesian?

Why are all these questions just an exercise in guessing what the assessor actually meant?

1. marty says:

Because the assessor is not sufficiently intelligent to compose a meaningful mathematical question, and they are not sufficiently practised in English to convey their semi-meaningful question in more than a semi-meaningful manner.

1. cathy says:

Do they lobotomize the Oompa-Loompas who write these questions before they let them loose on NAPLAN or do they only recruit really stupid ones to begin with?

1. marty says:

I think the latter, but working at ACARA must also be lobotomising.

1. Cathy says:

Think of all the PD you’d have to live through….

A whole virtual lobotomy in itself.

2. Dr. Mike says:

Cathy, I have doubts accessor knows the word Bayesian.

2. This question reminds me of the question about the condemned prisoner who is told that he will be hanged on one day during a Monday-Friday week, but will not know until it happens which day it is. What day will he be hanged? it cannot be Friday, because if he is not hanged on Thursday, he will know … etc.

To put a question like this on a state exam might not be any more sensible than the one supplied above, but it would be a lot more fun.

1. marty says:

Thanks, Ed. I think we can all identify with the condemned prisoner.

2. Back in Black says:

I was also reminded of this. But then I remembered that the prisoner, comfortable in the knowledge that they could not be hanged under the conditions of the sentence, was mighty surprised when the executioner turned up on Wednesday.

(The Paradox of the Unexpected Hanging).

3. JJ says:

So many of these remind me of Dame Slap in the Faraway Tree who posed these questions:
. If you take away three caterpillars from one bush, how many gooseberries will there be left ?
. If a train runs at six miles an hour and has to pass under four tunnels, put down what the guard’s mother is likely to have for dinner on Sundays?

It turned out that it didn’t matter what you answered, any answer was a first and Dame Slap was overjoyed and said she didn’t know.

Such a metaphor for bureaucracy (which is now so much a part of education). Brilliant stories.

4. Dr. Mike says:

🤣 Lucas buying lunch is a Markovian memoryless process 🤣

5. Christian R says:

My view on this question is not as negative as that of most other commenters. I stand to be corrected of course. Please excuse if the argument that follows appears to be a bit (or I fear, much) of an overshoot. My only issue with the question is stated in the last paragraph.

I came to think of the problem as a “non-rotated discrete” variant of Buffon’s needle problem. The latter problem is, I believe, standard fare of introductory probability courses, and there is a Wikipedia article on it (with much more information than we need here). But a few words on this first may help some readers.

Buffon’s needle problem talks about a needle thrown “at random” on an infinite set of parallel lines, and asks for the probability that it intersects, or “hits”, at least one of the lines. Assuming that the needle is shorter than the distance between any two closest parallel lines, as we shall do, it can hit at most one of those lines. The probability model that is inherent in the foregoing words “at random” is crucial in order to proceed. It “lives” on a two-dimensional space, one component of which is the distance of the mid-point of the needle to the nearest parallel line (which the needle may or may not intersect), the other the smaller of the two angles the needle makes with the parallel lines. We assume both of these random variables to be uniformly distributed, which is natural. Because we focus on the parallel line which is closest to the needle, the position of the mid-point of the needle is confined to a finite interval and the uniform distribution is thus no problem to specify. One further assumes that the said position and angle are independent random variables, which is also natural.

Now change – or, to a large extent, chop down – the Buffon probability model as follows. First, replace each parallel line with a hole at a fixed position on each line, such that the holes nicely line up for easier drawing or visualization. Then colour four adjacent holes blue, the next one green, then four blue, etc, in both directions, indefinitely. A green hole corresponds to a day when Lucas buys lunch. It is not important to the problem as we will set it up if the green hole is Monday, or Tuesday, …, or Friday as per what Lucas actually does. Finally, replace “today”, up to minor reformulation, by something like, “Inspector so-and-so visits Lucas on [or rather, after] a random school day to check if he has bought lunch.” The inspector is not a needle but a ball, has no orientation (this bit from Buffon’s needle problem is absent), and will occupy exactly one of the holes – which is the day he chooses to visit, or where he “drops” himself on what were the parallels in Buffon’s needle problem. He knows the date that he visits, giving him a location in absolute terms across the whole infinite space; but for our calculations, we don’t use the date, just the colours.

So in my opinion, the only substantial problem with the question – whose size I deem considerable but not outright horrible – is that “today” is indeed, as noted by others, not a very fortunate word here, and the randomness in choosing that day should have been at least alluded to in the question. The randomness in the inspector’s visiting day replaces the drunkenness that Marty mentioned in his above comment. If the inspector instead knows the lunch weekday, that is if he knows the colour of his chosen day, the model becomes trivial as others have noted. Another problem with the question, implicit from the above, may be that it requires a fair bit of work to make it rigorous while giving so little “meat”.

1. Back in Black says:

Hi CR, I think you’re correct in relating it to a buffoon’s needless problem.

2. marty says:

Thanks, Christian. I take your point. You’re wrong, but I take your point.

This is a probability question for Year 7 students. The first thing to note is that the question, in substance, is absolutely trivial. The only plausible answers are “Well, duh, 1/5.” and “How the hell would I know?”, the latter being preferable.

All that there is for a kid to do is either: (a) figure the question is trivial and go on, or; (b) start pondering the purpose and the meaning of “once every 5 school days” and “today”. Who is thinking about “today” and what that means? The surveyor, for whom it might be any old day but it happens to be “today”? Yeah, maybe. Not exactly the top of the list though.

The question is absolutely appalling, pointless and trivial and stupid and meaningless all wrapped together. The archetypal NAPLAN question.