Regardless of the irrelevant information in the question, the grammar leaves something to be desired, considering that the word “dice” refers to more than one die, leaving the question nonsensical.
Oh damnit.
Yes, indeed Luke Rhinehart.
I googled for the name of the author of The Dice Man, which I couldn’t remember, and not wearing my glasses as usual (because they make me feel old), read a preview of a result for Dice Men.
Is it better to be correct or to cling to delusions of youth?
What on earth are they trying to test? I guess the idea that the relative frequency is the best estimate of the probability, if we have no other information. When rolling a die there is some other information, but they try to dispel this by saying that the die may be biased.
Bayesian ideas may be too hard for Year 7, but are definitely too hard for these examiners.
Yeah, clearly information is unimportant in probability calculations.
I’d love to know how the writer of this question would calculate the probability of a false positive in a medical test …
Any kid who has ever experimented with a die will tell you that throwing 50 times isn’t enough to validate or even raise suspicion if the die is biased. Obviously, the question isn’t Bayesian. After all, it is year 7. This is terrible wording and, subsequently, a lousy question.
1 is all alone.
2 is a couple.
4 is too square.
5 is special (5 is the only prime number that has a sum of two consecutive prime numbers, namely 2 and 3).
6 is scared (because 7, 8, 9). (And 6 is too perfect).
I was making a serious point. Of all the numbers, the 3 total most closely approximates the expected number for an unbiased die. So, given the preamble, why pick on 3?
(This reminds me of the Physics Study Design a few cycles back where the magnitude of acceleration due to gravity was changed from 9.8 m/s^2 to 10 m/s^2, even though the students could use a scientific calculator in the exam. That madness did not survive subsequent Study Design changes, thank goodness).
I asked a teacher, who is much more experienced than I am in these matters, why NAPLAN uses the word “dice” rather than “die”. He thought that students are more likely to understand the question if the word “dice” were used.
I suppose that this is true. It would appear that the die is cast.
If we Chi-square test the results to find the probability of getting this result or something more extreme from a perfectly fair and balanced six-sided die, we get Chi^2 = 7.12 and p=0.21. 20%! One in every five times you performed this experiment of 50 rolls of a fair die, you would expect to get a result more extreme or equal to this! No sane statistician would accept that as statistically significant. In fact, a power analysis reveals you would need 339 rolls of a die with p(3) = 20% in order to rule out that null hypothesis with the traditional alpha=0.05.
But what is the right answer?? Surely we must incorporate both our prior belief that the die is probably fair, with our experimental evidence which leans towards some imbalance.
Taking a bayesian approach, the die can be represented by a dirichlet distribution with prior = (1/6, 1/6, 1/6, 1/6, 1/6, 1/6), with the strength of our prior belief represented by the number k of perfectly balanced rolls our prior belief is equal to. Our 50-roll experiment is has results of (6, 6, 10, 12, 4, 12), and our posterior probability of the next roll is given by (6+k, 6+k, 10+k, 12+k, 4+k, 12+k)/(6k+50). the expectation for the probability for rolling a 3 is given by (10 + k )/ (6k + 50), which can be taken anywhere from the prior of 1/6 (when k = infinity) to the frequentist 1/5 (when k -> 0) by scaling the strength of our prior k, while the 95% credible interval can cover anywhere from 1/10 to 1/3.
So clearly this NAPLAN question is testing a year 7 student’s understanding of frequentist and bayesian statistics.
I’m actually ok in principle with the question having kids assume that the experimental outcome is the best predictor of future rolls. But the question as framed is idiotic.
Regardless of the irrelevant information in the question, the grammar leaves something to be desired, considering that the word “dice” refers to more than one die, leaving the question nonsensical.
I agree. It is becoming more common to use “dice” to refer to a single die, but I always find it really jarring.
Right on, Libby!
Maybe in these current times the word die was deemed to have unpleasant connotations that might upset sensitive people.
Sorry, Marty. I think you missed a trick here. The obvious title is “Of Dice and Men.” (And another nod to Steinbeck – see https://mathematicalcrap.com/2022/11/09/witch-87-cannery-row/).
There is more than one novel title in my repertoire.
Aha. I see.
That was one weird novel (if you’re referring to The Dice Man by Ian Livingstone).
“Luke Rhinehart”.
Oh damnit.
Yes, indeed Luke Rhinehart.
I googled for the name of the author of The Dice Man, which I couldn’t remember, and not wearing my glasses as usual (because they make me feel old), read a preview of a result for Dice Men.
Is it better to be correct or to cling to delusions of youth?
Delusions of youth is preferable. But since I’m even beyond the delusions, I’m aiming for correctness.
I suppose you’re right. I’ve marginally better odds of being right about something than of being young again.
What on earth are they trying to test? I guess the idea that the relative frequency is the best estimate of the probability, if we have no other information. When rolling a die there is some other information, but they try to dispel this by saying that the die may be biased.
Bayesian ideas may be too hard for Year 7, but are definitely too hard for these examiners.
tom, the dice may be biased. Can’t you read?
Not a hill to die on.
Tom, you’re on a roll.
Yeah, clearly
information is unimportant in probability calculations.
I’d love to know how the writer of this question would calculate the probability of a false positive in a medical test …
Any kid who has ever experimented with a die will tell you that throwing 50 times isn’t enough to validate or even raise suspicion if the die is biased. Obviously, the question isn’t Bayesian. After all, it is year 7. This is terrible wording and, subsequently, a lousy question.
But even if they did, why pick on 3, of all numbers?
1 is all alone.
2 is a couple.
4 is too square.
5 is special (5 is the only prime number that has a sum of two consecutive prime numbers, namely 2 and 3).
6 is scared (because 7, 8, 9). (And 6 is too perfect).
3 is the obvious number to pick on.
I was making a serious point. Of all the numbers, the 3 total most closely approximates the expected number for an unbiased die. So, given the preamble, why pick on 3?
Fair point. I think they simply rolled the die to make their choice and 3 came up. (I’m serious).
Probably correct, simply thoughtlessness. But choosing 3 is is the idiotic cherry on top of a thoroughly idiotic question.
Yep, the quality of NAPLAN questions is a crapshoot in many ways.
Stop it with these jokes. They are not very funny, and they are distracting.
Apologies, but I did laugh each time.
Not about the test question.
3 gives a nice fraction 10/50 that is easy to convert into a percentage.
Yeah, because 4/50 is *way* too tough.
Particularly with a calculator.
(This reminds me of the Physics Study Design a few cycles back where the magnitude of acceleration due to gravity was changed from 9.8 m/s^2 to 10 m/s^2, even though the students could use a scientific calculator in the exam. That madness did not survive subsequent Study Design changes, thank goodness).
That madness also confused many students. They took it to be something like converting mm to cm instead of force to acceleration
I asked a teacher, who is much more experienced than I am in these matters, why NAPLAN uses the word “dice” rather than “die”. He thought that students are more likely to understand the question if the word “dice” were used.
I suppose that this is true. It would appear that the die is cast.
For Christ’s sake. They have a picture of a goddam die.
No more of these fucking jokes. You guys may like them, or think they are no big deal. But I’m sick of them, and I already indicated I’m sick of them.
Humour is fine, in moderation. Cheap fucking puns are not.
A colleague assured me that the Oxford English Dictionary regards “a dice” as acceptable.
You mean acceptable like it’s acceptable to use one’s phone at the MSO?
Which Oxford English Dictionary?
I did not ask him about the edition of OED to which he was referring.
It matters.
If we Chi-square test the results to find the probability of getting this result or something more extreme from a perfectly fair and balanced six-sided die, we get Chi^2 = 7.12 and p=0.21. 20%! One in every five times you performed this experiment of 50 rolls of a fair die, you would expect to get a result more extreme or equal to this! No sane statistician would accept that as statistically significant. In fact, a power analysis reveals you would need 339 rolls of a die with p(3) = 20% in order to rule out that null hypothesis with the traditional alpha=0.05.
But what is the right answer?? Surely we must incorporate both our prior belief that the die is probably fair, with our experimental evidence which leans towards some imbalance.
Taking a bayesian approach, the die can be represented by a dirichlet distribution with prior = (1/6, 1/6, 1/6, 1/6, 1/6, 1/6), with the strength of our prior belief represented by the number k of perfectly balanced rolls our prior belief is equal to. Our 50-roll experiment is has results of (6, 6, 10, 12, 4, 12), and our posterior probability of the next roll is given by (6+k, 6+k, 10+k, 12+k, 4+k, 12+k)/(6k+50). the expectation for the probability for rolling a 3 is given by (10 + k )/ (6k + 50), which can be taken anywhere from the prior of 1/6 (when k = infinity) to the frequentist 1/5 (when k -> 0) by scaling the strength of our prior k, while the 95% credible interval can cover anywhere from 1/10 to 1/3.
So clearly this NAPLAN question is testing a year 7 student’s understanding of frequentist and bayesian statistics.
Very funny, and very interesting. Thanks, Gordon.
I’m actually ok in principle with the question having kids assume that the experimental outcome is the best predictor of future rolls. But the question as framed is idiotic.