H2 maths tutor needs help

last SG · 2010-05-13T08:56:20+00:00

Hi Mad Hat, Part (ii) is alright. With a smaller x, the semicircle portion will look like a major arc instead. Thanks.Cheers, Wen ShihI believe you meant "y...

eagle SG

Singapore Asian Publications

wee_ws SG

Dear Mad Hat,

Thanks for being meticulous in checking for typos. Your answer of 0.785 is correct. You may send the list of errors to Dyna Publisher via email for their correction. In the past, I have identified errors in the answers too. You may also send the list to me via PM, so that I could put it up on my website for users of TYS. Acknowledgement will be given :)

Thanks again.

P.S. It is always good for us to teach students how to check their answers, like Eagle's reasonableness check.

Cheers,
Wen Shih

Mad Hat SG

Hi Mr. Wee, thanks for confirming that question! I have another similar question from the TYS which I will post soon. As for checking for typos, you're welcome. I will PM you a list before long - I'll check the latest DYNA edition first in case some of the typos have already been corrected.

In appreciation for all the help, may I share with everyone the following probability problems, which are simple yet enlightening:

Mr. Lim has two children. At least one of them is a boy. What is the probability that the other one is also a boy?

This is a famous trick question in conditional probability. The natural answer is 1/2 but, surprisingly, the correct answer is 1/3! This is within the H2 syllabus.

But now consider this question:

Mr. Tan has two children. You see him walking with a boy, which, he explains, is his son. (So now you know that at least one of his children is a boy.) What is the probability that the other one is also a boy?

This looks like the same question but, surprisingly, in this case, the answer of 1/2 is perfectly correct! Am I right?

Thankfully, the difference between the two questions is not within the H2 syllabus, but I'll be happy to discuss both problems if anyone is interested.

Lee012lee SG

Question 1

Mr. Lim has two children. At least one of them is a boy. What is the probability that the other one is also a boy?

Steps

(1) Given the condition at least one boy (BB, BG, GB) ie (1/2 x 1/2) x 3

(2) So, given the condition that at least one boy, the probability that the other one

is also a boy ie

P(BB)

= -----------------------

P(BB, BG, GB)

1/2 x 1/2

= ----------------------

( 1/2 x 1/2) x 3

= 1/3

Question 2

Steps

(1) Given the condition that the first child is already known as a boy ie a

probability of 1

(2) So, given that the condition that the first child is already known as a boy,

the probability that the other boy is also a boy

= 1 x 1/2 (alternatively (1/2) / 1 = 1/2)

= 1/2

PS : By the way, the correct answers for the H2 Maths past years examination questions can be found in the mark schemes provided by the Cambridge Chief examiner and these mark schemes and examiners' reports are sent to the heads of Maths departments of all the JCs and MI in Singapore.

wee_ws SG

Hi,

When I was still teaching in a school, our HOD would pass around the examiners' report for teachers' reference on common mistakes/misconceptions/answers. However, we never had access to the mark scheme from MOE. I wonder if the practice has changed, since I wrote to SEAB, years ago, to give teachers and parents access to mark schemes to increase transparency? This is a standard practice in UK and Australia.

Thanks.

Cheers,
Wen Shih

wee_ws SG

Hi,

Questions on statistics are usually tricky because of the play of words. Students will do better if we coach them on skills to interpret the meaning of sentences correctly. My students give me the impression that schools focus only on content rather than metacognitive strategies. Thanks.

Cheers,
Wen Shih

eagle SG

Actually I really wonder about how schools teach maths. It seems that they have the ability to turn simple concepts into amazingly difficult ones for the students. It doesn't help when students have self-defeating mentalities that it is hard as well.

Btw, sanity checks (or reasonableness checks) are common place in the field of engineering. Guess I was already part engineer mindset during my JC days with all these quick checks

Mad Hat SG

That's quite enlightening about the engineering mind-set.

I indeed had a bad lecturer in JC who made everything seem harder than it was. My current student also suffers because her teacher is too absorbed with complex formulae. When she shows me her lecture notes, I often cringe. They are overloaded with profound-looking formulae which intimidates students unnecessarily!

Is there a movement in Singapore devoted to improving the teaching of H2 maths?

Also, speaking of stats wording, can help me out with this, which I encountered this week? This is on my student's tutorial and supposedly from J79/P2/13 (Modified):

In a certain population, the heights of men are normally distributed with mean 172cm and standard deviation 10cm, and the heights of women are normally distributed with mean 165cm and standard deviation 8cm.

Calculate the probability that the height of a randomly chosen man exceeds that of a randomly chosen woman by 10cm.

We are given M ~ N(172, 10²) and W ~ N(165, 8²).

Okay, so we can define a new variable X = M - W.
Then X ~ N(7, 164).

But what is the question asking? It seems to be asking for P(X = 10), but that cannot be, since the answer would just be 0.

Maybe it is really asking for P(X > 10), in which case the wording should have been this:

Calculate the probability that the height of a randomly chosen man exceeds that of a randomly chosen woman by more than 10cm.

But when I calculate the answer, I get 0.4075, whereas the given answer is 0.309. So I don't really know what this question is asking. Any ideas?

SBS261P SG

is your student from RVHS?

SBS261P SG

In appreciation for all the help, may I share with everyone the following probability problems, which are simple yet enlightening:

This is a famous trick question in conditional probability. The natural answer is 1/2 but, surprisingly, the correct answer is 1/3! This is within the H2 syllabus.

But now consider this question:

This looks like the same question but, surprisingly, in this case, the answer of 1/2 is perfectly correct! Am I right?

Thankfully, the difference between the two questions is not within the H2 syllabus, but I'll be happy to discuss both problems if anyone is interested.

this is actually a re-incarnation of the famous Monty Hall problem.

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Forbiddensinner SG

this is actually a re-incarnation of the famous Monty Hall problem.

If it was me, I will keep with No. 1.

It seems as if the host is trying to play mind games with me and make me change my choice.

SBS261P SG

If it was me, I will keep with No. 1.

It seems as if the host is trying to play mind games with me and make me change my choice.

no, by keeping to your choice you have only a 1/3 chance of winning the car.

wee_ws SG

Dear Mad Hat,

We can always start a movement to help students understand H2 maths on this forum :)

Sometimes, simple things are made complex because of a fixated mindset that concepts must be taught in a certain way, rather than by the use of skilful means that can be understood at the level of the student :P

The original question has 4 parts, which are more meaningful in its assessment of concepts and skills. Calculate the probability that

(a) a man chosen at random is taller than 180 cm,

(b) a man and a woman chosen at random are both taller than 180 cm,

(d) a man chosen at random is taller than a woman chosen at random.

Sometimes, you can refuse to look at questions from school tutorials when they are not well-set and clear in their assessment outcomes.

Thanks.

Cheers,
Wen Shih

Lee012lee SG

Hi Mr Wee,

Indeed, I agree with you.

Badly set questions should be ignored.

A properly set question will be phrased in a very clear and precise language as those questions in the actual H2 Maths examination papers.

Like the earlier question on probability question on at least one boy, the setter tries purposely to trick students who attempt the question unnecessarily.

A more useful example in statistics could be a question to see whether the students know when to use the correct concept eg to use n x sigma^2 or n^2 x sigma^2 instead of trying to trick the students with the play of words.

Regards.

absol SG

Question 1

Mr. Lim has two children. At least one of them is a boy. What is the probability that the other one is also a boy?

Steps

(1) Given the condition at least one boy (BB, BG, GB) ie (1/2 x 1/2) x 3

(2) So, given the condition that at least one boy, the probability that the other one

is also a boy ie

P(BB)

= -----------------------

P(BB, BG, GB)

1/2 x 1/2

= ----------------------

( 1/2 x 1/2) x 3

= 1/3

hello, ive always been intriguied by this question ever since i saw a similar question on the wiki page of marylin vos savant. lol. i want to ask, in the above working, why would order of boy or girl matter?

Mad Hat SG

Dear Mad Hat,

We can always start a movement to help students understand H2 maths on this forum :)

The original question has 4 parts, which are more meaningful in its assessment of concepts and skills. Calculate the probability that

(a) a man chosen at random is taller than 180 cm,

(b) a man and a woman chosen at random are both taller than 180 cm,

(d) a man chosen at random is taller than a woman chosen at random.

Sometimes, you can refuse to look at questions from school tutorials when they are not well-set and clear in their assessment outcomes.

Thanks.

Cheers,
Wen Shih

Oh, I agree with this. I was just curious what the teacher was thinking such that he could come up with the answer 0.309. What question did he mean to set? But maybe you're right - it's not worth the time. Thank you for showing the original question. I was curious what modifications the teacher made.

As for the question absol just asked above, I think that's a good question, but I'll let Lee answer it!

Lee012lee SG

Hi Mad Hat,

The conditional probability question is a very simple conditional probability question.

Steps

(1) Draw the tree diagram

(2) Given condition at least one boy ie BB, BG, GB

(3) Require the other child will also be the boy ie BB

P(BB)

(4) Required Probability = --------------------

P(BB, BG, GB)

ie what is the probability of getting BB out of the given condition of (BB, BG,

GB).

PS : Thes types of conditional probability questions appear frequently in LCCI Business Statistics examination questions.

Mad Hat SG

LCCI Business Stats. Wah, I’d be interested to see what kinds of stats questions they ask on the exams.

Re SBS261P: my student is not from RV, but her friend is, so I have access to RV questions. And yes the question about the two boys is similar to the Monty Hall one, but I think the two boys question is 300 years old, whereas Monty Hall is only last century, so maybe Monty Hall is the reincarnation !

P.S. The Monty Hall problem can also be solved with conditional probability, if anyone wants to try. Need to think a bit how to do it, but it’s very satisfying to solve Monty Hall with conditional probability.

Lee012lee SG

LCCI Business Statistics exam question on conditional probability

A motor insurance company classifies its customers into

4 claims categories : Very Good (10%), Good(30%), Average (20%) and Poor (40%). The claims record shows that the percentage in each classification making at least one claim over the previous year.

Claims Category Percentage making at least one claim

Very Good 5%

Good 10%

Average 30%

Poor 60%

Find the probability that, if a customer is selected at random, the customer who makes at least one claim will come from the very good classification.

Mad Hat SG

LCCI Business Statistics exam question on conditional probability

A motor insurance company classifies its customers into

4 claims categories : Very Good (10%), Good(30%), Average (20%) and Poor (40%). The claims record shows that the percentage in each classification making at least one claim over the previous year.

Claims Category Percentage making at least one claim

Very Good 5%

Good 10%

Average 30%

Poor 60%

Find the probability that, if a customer is selected at random, the customer who makes at least one claim will come from the very good classification.

Hi Lee, thanks for sharing. Quite interesting to see the practical orientation of the question.

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