### School level maths question

Posted:

**Wed Apr 04, 2018 5:19 pm**Hello everyone. What is the answer to question 2?

A group for contestants and lovers of the Channel 4 game show 'Countdown'.

http://c4countdown.co.uk/

Page **1** of **1**

Posted: **Wed Apr 04, 2018 5:19 pm**

Hello everyone. What is the answer to question 2?

Posted: **Wed Apr 04, 2018 5:29 pm**

I'd have to guess that they're going for A, given that the only evidence you have is empirical. With no assumptions to the contrary, I think you have to assume that the first 50 spins obey the law of large numbers.

EDIT: If Question 2 is problematic, isn't Question 1 problematic in exactly the same way?

EDIT: If Question 2 is problematic, isn't Question 1 problematic in exactly the same way?

Posted: **Wed Apr 04, 2018 5:49 pm**

I'll come back to question 2 later (to see if other people have stuff to say first), but I took "relative frequency" just to mean the frequency in the actual results rather than probability.Johnny Canuck wrote: ↑Wed Apr 04, 2018 5:29 pmEDIT: If Question 2 is problematic, isn't Question 1 problematic in exactly the same way?

Posted: **Wed Apr 04, 2018 7:02 pm**

There's no reason to assume that every number is equally likely, so really the answer you'd probably get the mark for is A unless it's one of those annoying questions that tricks you into thinking the graph is in any way relevant.

Posted: **Wed Apr 04, 2018 9:45 pm**

Question 1 is B

Question 2 is A

I don't get the problem...

Relative frequency is used as an unbiased estimator for probability.

Question 2 is A

I don't get the problem...

Relative frequency is used as an unbiased estimator for probability.

Posted: **Thu Apr 05, 2018 12:36 am**

The real issue here is 'school level'

Posted: **Thu Apr 05, 2018 5:06 pm**

For my answer I'll copy and paste what I put on Facebook*:

Hello again. Thank you for all your replies (so far). I thought I might now put my own thoughts on the matter. Obviously I'm just another guy with an opinion, so I'm not intending this to be any sort of "final word" on the matter. Most of my thoughts on the matter have probably been covered across the various posts, but here they are anyway.

First of all, the answer given in the answers section is A - 76. No explanation is given, but I think you can all see where this comes from, whether you agree with it or not.

Right - I think it is a poor question, and that there isn't one "true" answer, regardless of how obvious you think it might be what the question-setter is getting at. Rather than just thinking of this as a problem in a maths paper, think of it as something that's happening in real life. If you actually spun a spinner 50 times and got these results, what would you expect from 200 spins?

You might start off thinking that it's probably a fair spinner (where the expected result from 200 spins would be 40 2s), but you now have 50 spins where 2 has come up more than you'd expect. Is it statistically significant? I haven't worked this out, but I don't think that this actually matters that much. Unless you were 100% confident to start with that it was a fair spinner (or had some other reason to think that it wasn't specifically biased towards 2), then this counts as evidence in favour of a bias towards 2 - statistical significance isn't a magic line that should cause discontinuous thinking once you cross it. To get an exact answer you'd need to have exact prior probabilities for the different probability distributions. But assuming that:

a) you don't start out 100% confident that it is a fair spinner, and

b) you start out neutral regarding the specific numbers - i.e. you have no initial reason to suspect bias towards one number over any other

(and maybe some other basic background assumptions that would be pretty obvious)

then the answer is not 40. It is greater than 40. Any prior probabilities you come up with regarding fairness etc. will give an answer greater than 40 once you get the results from the initial 50 spins.

But then is the answer 76? Out of the first 50 spins, 2 has come up 19 times. So if we take this to be a fair representation of the probability distribution, then out of 200 spins, you'd expect 76 2s. Can can we assume that it is a fair representation?

Imagine if the spinner had been spun once and it had landed on a 2. What would you expect from 200 spins? If you're going purely by the distribution, then you'd expect 200 2s. But in real life, would you be 100% sure that it would land on a 2 every time? No. You wouldn't even think it was remotely likely.

So you can't just automatically rely on the distribution. Of course, 1 isn't a good sample size, whereas 50 is much better. But there isn't some point where you discontinuously go from not trusting the sample size at all (and presumably using an assumption of fairness) to fully trusting it and using its results wholesale. There would be a gradual build up of confidence in the results. And this again goes back to prior probabilities. If you start out with the assumptions I outlined above:

a) you don't start out 100% confident that it is a fair spinner, and

b) you start out neutral regarding the specific numbers - i.e. you have no initial reason to suspect bias towards one number over any other

then you would give a 20% chance of each possible result (the numbers 1 to 5) coming out for the first spin.

Your prior probability of it being a fair spinner might be quite low, but with the neutrality assumption, you'd still need quite a bit of data to be confident that it was biased towards any particular number - 2 in this case. And the chance that you'd give to it landing on a 2 would always lag behind the data from the distribution. As said, you don't discontinuously go from the neutrality assumption to fully trusting the data. As you did more and more trials (hundreds, thousands, millions) the chance that you give of it landing on a 2 using your prior probabilities along with the data would converge upon the statistics from the data, but at no point would they exactly match - you'd need an infinite number of trials for that.

So the answer is not 76. It is less than 76. Using the assumptions above, the answer must be greater than 40 and less than 76. You can't put an exact figure on it without more information about the likely bias of the spinner - do we live in a world where spinners are routinely made fairly, or unfairly? Who gave you this spinner? What were their motivations?

In conclusion, poor question, but I'm going with 40 < answer < 76.

*I apologise for giving Facebook precedence over c4c in this case, although it doesn't reflect my overall bias. More people replied on Facebook so I went there first. But generally c4c trumps Facebook. I did also put the initial post on Facebook first, but that was because I could then use the photo on Facebook to link to here.

Hello again. Thank you for all your replies (so far). I thought I might now put my own thoughts on the matter. Obviously I'm just another guy with an opinion, so I'm not intending this to be any sort of "final word" on the matter. Most of my thoughts on the matter have probably been covered across the various posts, but here they are anyway.

First of all, the answer given in the answers section is A - 76. No explanation is given, but I think you can all see where this comes from, whether you agree with it or not.

Right - I think it is a poor question, and that there isn't one "true" answer, regardless of how obvious you think it might be what the question-setter is getting at. Rather than just thinking of this as a problem in a maths paper, think of it as something that's happening in real life. If you actually spun a spinner 50 times and got these results, what would you expect from 200 spins?

You might start off thinking that it's probably a fair spinner (where the expected result from 200 spins would be 40 2s), but you now have 50 spins where 2 has come up more than you'd expect. Is it statistically significant? I haven't worked this out, but I don't think that this actually matters that much. Unless you were 100% confident to start with that it was a fair spinner (or had some other reason to think that it wasn't specifically biased towards 2), then this counts as evidence in favour of a bias towards 2 - statistical significance isn't a magic line that should cause discontinuous thinking once you cross it. To get an exact answer you'd need to have exact prior probabilities for the different probability distributions. But assuming that:

a) you don't start out 100% confident that it is a fair spinner, and

b) you start out neutral regarding the specific numbers - i.e. you have no initial reason to suspect bias towards one number over any other

(and maybe some other basic background assumptions that would be pretty obvious)

then the answer is not 40. It is greater than 40. Any prior probabilities you come up with regarding fairness etc. will give an answer greater than 40 once you get the results from the initial 50 spins.

But then is the answer 76? Out of the first 50 spins, 2 has come up 19 times. So if we take this to be a fair representation of the probability distribution, then out of 200 spins, you'd expect 76 2s. Can can we assume that it is a fair representation?

Imagine if the spinner had been spun once and it had landed on a 2. What would you expect from 200 spins? If you're going purely by the distribution, then you'd expect 200 2s. But in real life, would you be 100% sure that it would land on a 2 every time? No. You wouldn't even think it was remotely likely.

So you can't just automatically rely on the distribution. Of course, 1 isn't a good sample size, whereas 50 is much better. But there isn't some point where you discontinuously go from not trusting the sample size at all (and presumably using an assumption of fairness) to fully trusting it and using its results wholesale. There would be a gradual build up of confidence in the results. And this again goes back to prior probabilities. If you start out with the assumptions I outlined above:

a) you don't start out 100% confident that it is a fair spinner, and

b) you start out neutral regarding the specific numbers - i.e. you have no initial reason to suspect bias towards one number over any other

then you would give a 20% chance of each possible result (the numbers 1 to 5) coming out for the first spin.

Your prior probability of it being a fair spinner might be quite low, but with the neutrality assumption, you'd still need quite a bit of data to be confident that it was biased towards any particular number - 2 in this case. And the chance that you'd give to it landing on a 2 would always lag behind the data from the distribution. As said, you don't discontinuously go from the neutrality assumption to fully trusting the data. As you did more and more trials (hundreds, thousands, millions) the chance that you give of it landing on a 2 using your prior probabilities along with the data would converge upon the statistics from the data, but at no point would they exactly match - you'd need an infinite number of trials for that.

So the answer is not 76. It is less than 76. Using the assumptions above, the answer must be greater than 40 and less than 76. You can't put an exact figure on it without more information about the likely bias of the spinner - do we live in a world where spinners are routinely made fairly, or unfairly? Who gave you this spinner? What were their motivations?

In conclusion, poor question, but I'm going with 40 < answer < 76.

*I apologise for giving Facebook precedence over c4c in this case, although it doesn't reflect my overall bias. More people replied on Facebook so I went there first. But generally c4c trumps Facebook. I did also put the initial post on Facebook first, but that was because I could then use the photo on Facebook to link to here.

Posted: **Thu Apr 05, 2018 6:59 pm**

I agree that it’s a poorly worded question but disagree with the 40<n<76 bit.

The question should probably be ‘based on these results, how many would you expect to get etc.’

The word expect though I think probably covers up your problem with the question.

The reasons I don’t like ‘40<n<76’:

Whilst I don’t think it should be assumed the spinner isn’t fair, nor should we think it is fair. For this end, using 40 as any sort of bound to a range is fairly arbitrary.

The results from they first 50 spins suggest it isn’t fair. By your logic, we don’t know this is the case, but why use 40 as a lower bound? Surely your response should be ‘0<n<200’?

Although it’s the expectation part which I think you’re skimming over.

If you really wanted to do it properly, I would suggest using a z-test to get a confidence interval. (Or else a t-test with 49 degrees of freedom).

To be honest though, you’ve pretty much elucidated the problems I have with most statistical analysis. It’s just that, analysis. Too subjective for my liking.

The question should probably be ‘based on these results, how many would you expect to get etc.’

The word expect though I think probably covers up your problem with the question.

The reasons I don’t like ‘40<n<76’:

Whilst I don’t think it should be assumed the spinner isn’t fair, nor should we think it is fair. For this end, using 40 as any sort of bound to a range is fairly arbitrary.

The results from they first 50 spins suggest it isn’t fair. By your logic, we don’t know this is the case, but why use 40 as a lower bound? Surely your response should be ‘0<n<200’?

Although it’s the expectation part which I think you’re skimming over.

If you really wanted to do it properly, I would suggest using a z-test to get a confidence interval. (Or else a t-test with 49 degrees of freedom).

To be honest though, you’ve pretty much elucidated the problems I have with most statistical analysis. It’s just that, analysis. Too subjective for my liking.

Posted: **Thu Apr 05, 2018 8:46 pm**

I'm fully with Noel on this matter, there's no reason to say any answer other than 76 really based on the data. Obviously it's possible this just happened by chance and it is a fair spinner, just as you could toss a fair coin 5 times and it could land on it's edge each time. With nothing else to go on though, you have to assume the data is representative and the answer is therefore 76.

Posted: **Thu Apr 05, 2018 8:53 pm**

Very interesting conclusion, why specifically less than 76? Can it not BE 76, it could even be MORE than 76 theoretically. The question isn't "create a 90% confidence interval for where the number of 2s in 200 throws would lie" or whatever. It specifically asks about EXPECTED value. If statistics showed that someone averaged 11 maxes a game on apterous over their last 50 games, is it sensible to then conclude that for their next games you'd expect them to average less than 11? Off a large(ish) sample size, surely you'd expect the average to be very close to the average found through statistics (obvs assuming no improvement for arguments sake). You'd expect them to average between 10.5 and 11.5, but if asked for a specific value why would you not say 11?Gavin Chipper wrote: ↑Thu Apr 05, 2018 5:06 pmFor my answer I'll copy and paste what I put on Facebook*:

blah blah blah blah

So the answer is not 76. It is less than 76. Using the assumptions above, the answer must be greater than 40 and less than 76. You can't put an exact figure on it without more information about the likely bias of the spinner - do we live in a world where spinners are routinely made fairly, or unfairly? Who gave you this spinner? What were their motivations?

blah blah blah blah

Posted: **Thu Apr 05, 2018 10:02 pm**

The reason I'm using 40 as a lower bound is that I started off with no assumption about the spinner favouring one particular number over another. The two main assumptions again:Noel Mc wrote: ↑Thu Apr 05, 2018 6:59 pmI agree that it’s a poorly worded question but disagree with the 40<n<76 bit.

The question should probably be ‘based on these results, how many would you expect to get etc.’

The word expect though I think probably covers up your problem with the question.

The reasons I don’t like ‘40<n<76’:

Whilst I don’t think it should be assumed the spinner isn’t fair, nor should we think it is fair. For this end, using 40 as any sort of bound to a range is fairly arbitrary.

The results from they first 50 spins suggest it isn’t fair. By your logic, we don’t know this is the case, but why use 40 as a lower bound? Surely your response should be ‘0<n<200’?

Although it’s the expectation part which I think you’re skimming over.

If you really wanted to do it properly, I would suggest using a z-test to get a confidence interval. (Or else a t-test with 49 degrees of freedom).

To be honest though, you’ve pretty much elucidated the problems I have with most statistical analysis. It’s just that, analysis. Too subjective for my liking.

a) you don't start out 100% confident that it is a fair spinner, and

b) you start out neutral regarding the specific numbers - i.e. you have no initial reason to suspect bias towards one number over any other

So yes, it might be that in the real world, most spinners are designed with a bias against the number 2 (though I'd imagine this is unlikely), and taking this into account with my prior probabilities, I might still end up with an answer lower than 40, despite the disproportionate number of 2s that have come out. But given the above assumption on neutrality, the greater than average number of 2s can only shift my expectation upwards from 40.

To be clear, I'm not saying that the spinner is definitely biased in favour of 2 and that with a large number of spins, we're guaranteed to get >20% 2s. What I'm saying is that based on our knowledge of the situation, our expectation is that there will be >20% 2s. Our expectation and the expectation of someone that has fully analysed the spinner might be different.

To give another example:

With probability 50% you are given a fair coin. With probability 50% you are given a coin that always lands on heads. How many heads would you expect to get after 10 tosses of this coin? The answer is 7.5, but that doesn't mean that the coin is definitely biased.

Same as above. I'm not talking about the "objective" expected value, but the expected value based on the information that we have. And think of the case where someone has averaged 15 maxes a game or 0 maxes a game.Elliott Mellor wrote: ↑Thu Apr 05, 2018 8:53 pmVery interesting conclusion, why specifically less than 76? Can it not BE 76, it could even be MORE than 76 theoretically. The question isn't "create a 90% confidence interval for where the number of 2s in 200 throws would lie" or whatever. It specifically asks about EXPECTED value. If statistics showed that someone averaged 11 maxes a game on apterous over their last 50 games, is it sensible to then conclude that for their next games you'd expect them to average less than 11? Off a large(ish) sample size, surely you'd expect the average to be very close to the average found through statistics (obvs assuming no improvement for arguments sake). You'd expect them to average between 10.5 and 11.5, but if asked for a specific value why would you not say 11?Gavin Chipper wrote: ↑Thu Apr 05, 2018 5:06 pmFor my answer I'll copy and paste what I put on Facebook*:

blah blah blah blah

So the answer is not 76. It is less than 76. Using the assumptions above, the answer must be greater than 40 and less than 76. You can't put an exact figure on it without more information about the likely bias of the spinner - do we live in a world where spinners are routinely made fairly, or unfairly? Who gave you this spinner? What were their motivations?

blah blah blah blah

Edit - just to add to this bit, you could look at the stats over all the years of Apterous to get a distribution of what proportion of people get what average number of maxes (obviously stuff like improvement and dictionary changes make it more complex in real life but we'll ignore that for now). Someone averaging 11 maxes over 50 games could be an 11.5-maxer slightly underperforming, a 10.5-maxer slightly overperforming etc. But we could put all the numbers in and work out which was more likely. If there are more 10.5-maxers than 11.5-maxers, for example, they are more likely to come from the 10.5-maxer group. And that is what brings our expected value of their overall max rate down below 11.

Posted: **Fri Apr 06, 2018 6:25 pm**

The real problem is that there is so much debate over 2 marks in total...in an exam I'd just circle 0.38, 76 and be done with it. The only reason people are seeing problems is the context, had it been something completely different there'd be no question it was 76.