The Mathematics Thread

[anat0l]

3-week-old story now, but still intrigues me.

This question was set for a UK GCSE maths exam. It started trending on Twitter and sprouted an online petition with a few thousand signatories, because in the students' opinion, it was inappropriately difficult for the level of exam.

I solved it in mere minutes, and I haven't really done probability problems for ages. I'm lucky to remember how to factor quadratics (although that skill is not required by the question).

What do you think? Any of you have children aged between 15-18 years who are studying maths, should probably pass this to them to see if they can do it.

There are [I]n[/I] sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow.

Hannah takes a random sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3.

Show that [I]n[/I]² - [I]n[/I] - 90 = 0

 

[JohnK]

Re: The totally off-topic thread

3-week-old story now, but still intrigues me.

This question was set for a UK GCSE maths exam. It started trending on Twitter and sprouted an online petition with a few thousand signatories, because in the students' opinion, it was inappropriately difficult for the level of exam.

I solved it in mere minutes, and I haven't really done probability problems for ages. I'm lucky to remember how to factor quadratics (although that skill is not required by the question).

What do you think? Any of you have children aged between 15-18 years who are studying maths, should probably pass this to them to see if they can do it.

There are [I]n[/I] sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow.

Hannah takes a random sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3.

Show that [I]n[/I]² - [I]n[/I] - 90 = 0

I don't even remember probability. Working out n was simple. I had that answer before finishing reading. Is there more to the question?

 

[medhead]

Re: The totally off-topic thread

3-week-old story now, but still intrigues me.

This question was set for a UK GCSE maths exam. It started trending on Twitter and sprouted an online petition with a few thousand signatories, because in the students' opinion, it was inappropriately difficult for the level of exam.

I solved it in mere minutes, and I haven't really done probability problems for ages. I'm lucky to remember how to factor quadratics (although that skill is not required by the question).

What do you think? Any of you have children aged between 15-18 years who are studying maths, should probably pass this to them to see if they can do it.

There are [I]n[/I] sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow.

Hannah takes a random sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3.

Show that [I]n[/I]² - [I]n[/I] - 90 = 0

As per my reply on Facebook, it's actually incredibly easy if you have just a basic understanding of probability. This makes me question how this is even an extension question to separate an A from a C. Surely it should be more like a C from an F.

Having said that I've had someone with the same level of high school maths as me (even the same classes) express the view that powerball had a higher probability of winning (easier to win) that the usual 6 from 45 Saturday night lotto. Perhaps that is the difference with university stats and probability, beyond 1st year, and the high school version.

 

[medhead]

Re: The totally off-topic thread

I don't even remember probability. Working out n was simple. I had that answer before finishing reading. Is there more to the question?

The question doesn't ask you to work out n, it asks to prove n^2-n-90=0. That's a slightly different proposition. You need to prove that equation for all possible values of n not just one.

 

[juddles]

Re: The totally off-topic thread

The question doesn't ask you to work out n, it asks to prove n^2-n-90=0. That's a slightly different proposition. You need to prove that equation for all possible values of n not just one.

medhead, although you are mainly correct, as usual you are also somewhat incorrect.

yes, the point of the exercise is not to calculate the value of n

but neither is there any requirement to prove the equation for "all possible values of n", you just need to show why the probability is 1/3 if n2 - n - 90 = 0

 

[RAM]

Re: The totally off-topic thread

I wonder what the chances are of that possibility regarding probability?

Of course the answer is...



42.

 

[medhead]

Re: The totally off-topic thread

medhead, although you are mainly correct, as usual you are also somewhat incorrect.

yes, the point of the exercise is not to calculate the value of n

but neither is there any requirement to prove the equation for "all possible values of n", you just need to show why the probability is 1/3 if n2 - n - 90 = 0

You've got that around the wrong way, 1/3 is a given, you don't need to show why the probability is 1/3. It is a general proof for all possible values of n when the overall probability is 1/3.

If the overall probability was 1/4 then it would be prove n^2-n-120=0

 

[juddles]

Re: The totally off-topic thread

You've got that around the wrong way, 1/3 is a given, you don't need to show why the probability is 1/3. It is a general proof for all possible values of n when the overall probability is 1/3.

If the overall probability was 1/4 then it would be prove n^2-n-120=0

Am disagreeing, but instead of debating this, lets play this traditionally - show me your solution (workings) :

Edit: IE show me what you would have written to answer the actual question

 

[medhead]

Re: The totally off-topic thread

Am disagreeing, but instead of debating this, lets play this traditionally - show me your solution (workings) :

Edit: IE show me what you would have written to answer the actual question

P(draw first orange) =6/n
P(draw second orange) =5/(n-1)
P(drawing 2 consecutive oranges) =6/n * 5/(n-1)=1/3
(6*5)/n(n-1)=1/3
n(n-1)=6*5*3=90
N^2-n-90=0

It only works when the overall probability is 1/3.

If we factorise n^2-n-90 we get (n-10)(n+9). Solutions for n are 10, or -9 (obviously a non real solution)

 

[david1]

Re: The totally off-topic thread

You can't start with n^2-n-90=0, that's what you are trying to show. You have to start with the probability being 1/3rd, and combine that with the probability of getting 2 oranges in a row starting with n items. It's no different to make the probability P and work it out in general.

On a related note, as a maths PhD student, I mark a lot of maths questions, and the number of times use the result that they are trying to prove in their proof is amazing!

@RAM why are consecutive spins at one wheel not independent but spins at different wheels are? And since the RTP isn't the probability of winning but the average return, and since not all payouts are the same size, I don't think you can't directly compare RTP with probability of winning. So I'm not too sure about your method.

(And medhead, -9 is most definitely a real number!)

 

[serfty]

Re: The totally off-topic thread

Real numbers do not always equate to real solutions.

You can have 10 gum drops in your possession ... you may have 0 ... but you can't physically possess -9 (although one might owe 9 gumdrops to a gumdrop loan shark)

 

[david1]

Re: The totally off-topic thread

Haha I was only playing! -9 is just as real a solution to the equation as 10, but it's a non-physical solution that doesn't solve the actual problem.

I do sometimes get over zealous about this kind of thing, often when marking and having to put up with how lazy students are with precise definitions!!

 

[medhead]

Re: The totally off-topic thread

You have to start with the probability being 1/3rd, and combine that with the probability of getting 2 oranges in a row starting with n items. It's no different to make the probability P and work it out in general.

Pretty sure that's what I did do.

On a related note, as a maths PhD student, I mark a lot of maths questions, and the number of times use the result that they are trying to prove in their proof is amazing!

I won't hold being a maths student against you.

(And medhead, -9 is most definitely a real number!)

Lucky I didn't say it wasn't a real number, then.

It is a non real solution, impossible to have -9 items and then draw something out. Even if the magician's guild says otherwise.

 

[trixydoza]

Re: The totally off-topic thread

It's no different to make the probability P and work it out in general.

Non-trivial to come up with positive integer solutions for generalized P.

 

[SeatBackForward]

Re: The totally off-topic thread

Real numbers do not always equate to real solutions.

You can have 10 gum drops in your possession ... you may have 0 ... but you can't physically possess -9 (although one might owe 9 gumdrops to a gumdrop loan shark)

Isn't that a short position on Gumdrops?

 

[david1]

Re: The totally off-topic thread

Pretty sure that's what I did do.

Yeah, I didn't say you didn't! Was responding to the discussion about that issue.

Non-trivial to come up with positive integer solutions for generalized P.

Good point! If P=120/(m^2-1), for m odd and greater than or equal to 11, then n is an integer.

 

[anat0l]

Re: The totally off-topic thread

Seems the question was way too easy, so what those poor kids were crowing about I have absolutely no idea. Getting part marks for this question, if that is applicable, would still be very possible even if a student could not answer the whole thing. That is not markedly difficult by a long shot.

If they thought previous years of practice exams were markedly easier than this, then that is just odd as.

On a semi-related note, as someone who has both relied on past exams and then had to recommend questions to be set for exams, I do share some concern about setting similar questions year-in-year-out, along with issues of grade creep.

You can't start with n^2-n-90=0, that's what you are trying to show. You have to start with the probability being 1/3rd, and combine that with the probability of getting 2 oranges in a row starting with n items. It's no different to make the probability P and work it out in general.

This, or much of mathematical proofs, does my head in. Deductive reasoning, fine, however we know that is not the only way to prove something in mathematics. You can also use reductio ad absurdum, but in this case it will not work. The one that really does me in is proof by induction, which actually supposes the final conclusion (or what is to be proven) to be true, and works backwards or directly substitutes into the application in order to prove it works, but not necessarily show that it was true in the first place.

However, it induction is a legitimate form of mathematical proof, then I see no reason why starting with the quadratic and solving for it to find n, then developing a probability tree to demonstrate the 1/3 chance to show this value of n works, is not an unreasonable approach. One might argue, fine, for your working you do this, but then when you write your answer, you need to write it down in the reverse (to make it look like it was in fact a deductive approach). I think the question is rather silly because showing the relationship like this merely encourages having to solve for n anyway; it would have been simply easier to ask the student to develop an equation to determine n, then solve for it, or simply just find the value of n (but add an additional note so you can discard unintelligible attempts like random trial-and-error).

Of course, if n had no real (or positive, or more precisely, positive integer) solutions, the "solve first" approach would fall down significantly, because you can't automatically assume that a 'non-sensical' n means the situation is impossible, since the question was to show that n is the number of sweets in the bag, not what n actually is.

(And medhead, -9 is most definitely a real number!)

Semantics of words can have unintended misunderstandings between mathematics and the rest of the world!

The whole (pun intended) thing is rather complex (pun intended)

 

[david1]

Re: The totally off-topic thread

Each machine has its own cpu monitoring the win rate (I forget the technical term). I decided to find out EXACTLY how the machines worked, so as a funds manager I got the opportunity to question the guy who started Aristocrat Leisure one day. He was very happy to go into the absolute detail at length.

So as long as each machine is not part of a linked jackpot system then consecutive spins on the same machine are not independent as the CPU is going to ensure the stipulated win rate is achieved over the play cycle. Some people would say the tempo of the music will even speed up the more you lose but they're cynics of course.

Wow that's interesting!

Seems the question was way too easy, so what those poor kids were crowing about I have absolutely no idea. Getting part marks for this question, if that is applicable, would still be very possible even if a student could not answer the whole thing. That is not markedly difficult by a long shot.

Yes, it's not a hard question, but I think the wording is what confuses people, at first glance it sounds like the n^2-n-90=0 comes out of nowhere, I think it makes people think of thing s like this: http://www.fbcomics.com/images/comics/Maths in class and EXAM.jpg

This, or much of mathematical proofs, does my head in. Deductive reasoning, fine, however we know that is not the only way to prove something in mathematics. You can also use reductio ad absurdum, but in this case it will not work. The one that really does me in is proof by induction, which actually supposes the final conclusion (or what is to be proven) to be true, and works backwards or directly substitutes into the application in order to prove it works, but not necessarily show that it was true in the first place.

However, it induction is a legitimate form of mathematical proof, then I see no reason why starting with the quadratic and solving for it to find n, then developing a probability tree to demonstrate the 1/3 chance to show this value of n works, is not an unreasonable approach. One might argue, fine, for your working you do this, but then when you write your answer, you need to write it down in the reverse (to make it look like it was in fact a deductive approach). I think the question is rather silly because showing the relationship like this merely encourages having to solve for n anyway; it would have been simply easier to ask the student to develop an equation to determine n, then solve for it, or simply just find the value of n (but add an additional note so you can discard unintelligible attempts like random trial-and-error).

Of course, if n had no real (or positive, or more precisely, positive integer) solutions, the "solve first" approach would fall down significantly, because you can't automatically assume that a 'non-sensical' n means the situation is impossible, since the question was to show that n is the number of sweets in the bag, not what n actually is.

There's nothing to prove here, it's just calculating probabilities.

Induction doesn't work quite as you said either. To use induction you need a statement that is true for any k. To prove by induction you need to show a base case is true, say, it's true when k=1. You also need to prove that if it is true for k=m (for an arbitrary m), then it is also true for k=m+1. That shows that if it's true for k=1 then it is true for k=2. If it is true for k=2 it is true for k=3. And so on, up to any positive integer you can think of.

Anyway, I think that's enough from me!!

 

[anat0l]

Re: The totally off-topic thread

For what it is worth, here is a sample UK GCSE maths test from the BBC website (sub-site called "Bitesize"). This particular test has a "no calculator" restriction on it, but you do need a pair of compasses and a ruler.

http://downloads.bbc.co.uk/schools/gcsebitesize/maths/mocks/mathsmockh1_nocalc.pdf

From the final exam of grade 8, we had a calculator for every maths test thereafter. Of course, that doesn't mean everyone flew through the exams, it just means that you need to actually have problem solving skills. Also, a calculator won't help you reduce algebraic expressions or complete trigonometric proofs.

 

[medhead]

Re: The totally off-topic thread

Of course, if n had no real (or positive, or more precisely, positive integer) solutions, the "solve first" approach would fall down significantly, because you can't automatically assume that a 'non-sensical' n means the situation is impossible, since the question was to show that n is the number of sweets in the bag, not what n actually is.

In that question, in theory, n could be anything - even a non-integer. Being able to deduce n as 10 certainly helps along with a bit of quick mental arithmetic i.e. 5x6, and 5x6x3, to recognise the relationship with 90.

For what it is worth, here is a sample UK GCSE maths test from the BBC website (sub-site called "Bitesize"). This particular test has a "no calculator" restriction on it, but you do need a pair of compasses and a ruler.

http://downloads.bbc.co.uk/schools/gcsebitesize/maths/mocks/mathsmockh1_nocalc.pdf

From the final exam of grade 8, we had a calculator for every maths test thereafter. Of course, that doesn't mean everyone flew through the exams, it just means that you need to actually have problem solving skills. Also, a calculator won't help you reduce algebraic expressions or complete trigonometric proofs.

One from my daughter's year 9 maths geometry assignment: calculator got her an answer of 276 for 2 x Pi x 2.5 x 2. Test review made me explaining the concept of estimation Pi~3, so the answer has to be about 30. Therefore if your answer is 276 it must be wrong.