### Modal value & Expected value

Let us look at the difference between modal value and expected value. We shall start by saying they are different, albeit close.

Modal value refers to the mode, that is, the value that has the highest probability (chance) of occurring.

Expected value refers to the value, we expect to have, on average.

Before we start, I’ll do a fast recap on Binomial Distribution, $X \sim \text{B}(n, p)$ by flashing the formulae that we can find on MF26.

$\text{P}(X = x) = ^n C_x (p)^x (1-p)^{n-x}$

$\mathbb{E}(X) = np$

$\text{Var}(X) = np(1-p)$

The expected value is simply given by $\mathbb{E}(X)$.

Now to find the modal value, we have to go through a slightly nasty and long working. You may click and find out.

We have that $\frac{\text{P}(X = r + 1)}{\text{P}(X = r)} = \frac{(n-r)}{(r+1)} \frac{p}{1-p}$. This is what we call the recurrence formula. We consider this to give us the ratio between successive probabilities. And to illustrate how this works, nothing beats an example question.

Consider candies are packed in packets of 20. On average the proportion of candies that are blue-colored is $p$. It is know that the most common number of blue-colored candies in a packet is 6. Use this information to find exactly the range of values that $p$ can take.

First, most common number is the same as saying the modal/ highest frequency.

This means that $\text{P}(X=6)$ is the highest/ largest probability… Let us turn our attention to the recurrence formula now. If $\text{P}(X=6)$ is the largest, then it means that $\text{P}(X=6) \textgreater \text{P}(X=7)$ and also $\text{P}(X=6) \textgreater \text{P}(X=5)$.

Lets start by looking at the first one… $\text{P}(X=6) \textgreater \text{P}(X=7)$

$\text{P}(X=6) \textgreater \text{P}(X=7)$

$1 > \frac{\text{P}(X=7)}{\text{P}(X=6)}$

$\frac{\text{P}(X=7)}{\text{P}(X=6)} \textless 1$

But hold on! This looks like the recurrence formula. (ok, in exams, its either you use the recurrence formula or derive on the spot. Both works!)

Now I’ll advice you try the second one (before clicking on answer) on your own, that is, $\text{P}(X=6) > \text{P}(X=5)$.

Now, if the question simply says that the expected number of blue-colored candies in a packet of 20 is 6. Then

$\mathbb{E}(X) = 6$

$(20)p = 6$

$p = \frac{3}{10}$

We observe that this value actually falls in the range of $p$ we found.

### DRV questions with a twist

I want to share a question that is really old (older than me, actually). It is from the June 1972 paper. As most students know, Maclaurin’s series was tested in last year A’levels Paper 1 as a sum to infinity. And this DRV did the same thing. Here is the question.

A bag contains 6 black balls and 2 white balls. Balls are drawn at random one at a time from the bag without replacement, and a white ball is drawn for the first time at R th draw.

(i) Tabulate the probability distribution for R.

(ii) Show that E( R ) = 3, and find Var( R ).

If each ball is replaced before another is drawn, show that in this case E( R ) = 4.

### Solutions to Review 1

Question 1
(i)
$y = f(x) = \frac{x^2 + 14x + 50}{3(x+7)}$

$3y(x+7) = x^2 + 14x + 50$

$x^2 + (14-3y)x + 50 - 21 y = 0$

$\text{discriminant} \ge 0$

$(14-3y)^2 - 4(1)(50-21y) \ge 0$

$196 - 84y + 9y^2 - 200 + 84y \ge 0$

$9y^2 - 4 \ge 0$

$(3y - 2)(3y + 2) \ge 0$

$y \le - \frac{2}{3} \text{~or~} y \ge \frac{2}{3}$

(ii)
Using long division, we find that

$y = \frac{x^2 + 14x + 50}{3(x+7)} = \frac{x}{3} + \frac{7}{3} + \frac{1}{3(x+7)}$

So the asymptotes are $y = \frac{x}{3} + \frac{7}{3}$ and $x = -7$

Question 2
(i)
$x^2 - 9y^2 + 18y = 18$

$x^2 - 9(y^2 - 2y) = 18$

$x^2 - 9[(y-1)^2 - 1^2] = 18$

$x^2 - 9(y-1)^2 + 9 = 18$

$x^2 - 9(y-1)^2 = 9$

$\frac{x^2}{9} - (y-1)^2 = 1$

This is a hyperbola with centre $(0, 1)$, asymptotes are $y = \pm \frac{x}{3} + 1$, and vertices $(3, 1)$ and $(-3, 1)$.

$y = \frac{1}{x^2} + 1$ is a graph with asymptotes $x = 0$ and $y=1$.

Use GC to plot.

(ii)
$\frac{x^2}{9} - (y-1)^2 = 1$—(1)

$y = \frac{1}{x^2} + 1$ —(2)

Subst (2) to (1),

$\frac{x^2}{9} - (\frac{1}{x^2} + 1 - 1)^2 = 1$

$\frac{x^2}{9} - (\frac{1}{x^2})^2 = 1$

$x^2 - \frac{9}{x^4} = 9$

$x^6 - 9 = 9x^4$

$x^6 - 9x^4 - 9 = 0$

(iii)
From graph, we observe two intersections. Thus, two roots.

Question 3
(ai)
$\sum_{r=1}^n (r+1)(3r-1)$

$= \sum_{r=1}^n (3r^2 + 2r -1)$

$= \sum_{r=1}^n 3r^2 + \sum_{r=1}^n 2r - \sum_{r=1}^n 1$

$= 3 \sum_{r=1}^n r^2 + 2 \sum_{r=1}^n r - \sum_{r=1}^n 1$

$= 3 \frac{n}{6}(n+1)(2n+1) + 2 \frac{n}{2}(1 + n) - n$

$= \frac{n}{2}(n+1)(2n+1) + n(1+n) - n$

$= \frac{n}{2}(n+1)(2n+1) + n^2$

(aii)
$2 \times 4 + 3 \times 10 + 4 \times 16 + ... + 21 \times 118$

$= 2 [2 \times 2 + 3 \times 5 + 4 \times 8 + ... + 21 \times 59]$

$= 2 [(1+1) \times (3 \cdot 1 - 1) + (2+1) \times (3 \cdot 2 -1) + (3+1) \times (3 \cdot 3 -1) + ... + (20+1) \times (3 \cdot 20 -1) ]$

$= 2 \sum_{r=1}^{20} (r+1)(3r-1)$

$= 2 [\frac{n}{2}(n+1)(2n+1) + n^2 ]$

$= n(n+1)(2n+1) + n^2$

$= n(2n^2 + 3n + 1) + n^2$

$= 2n^3 + 4n^2 + n$

(bi)
$\frac{2}{(r-1)(r+1)} = \frac{A}{r-1} - \frac{B}{r+1}$

$2 = A(r+1) - B(r-1)$

Let $r = -1$

$2 = - B(-2) \Rightarrow B = 1$

Let $r = 1$

$2 = A(2) \Rightarrow A = 1$

$\therefore \frac{2}{(r-1)(r+1)} = \frac{1}{r-1} - \frac{1}{r+1}$

(bii)
$\sum_{r=2}^n \frac{1}{(r-1)(r+1)}$

$= \frac{1}{2} \sum_{r=2}^n \frac{2}{(r-1)(r+1)}$

$= \frac{1}{2} \sum_{r=2}^n (\frac{1}{r-1} - \frac{1}{r+1})$

$= \frac{1}{2} [ 1 - \frac{1}{3}$

$+ \frac{1}{2} - \frac{1}{4}$

$+ \frac{1}{3} - \frac{1}{5}$

$...$

$+ \frac{1}{n-3} - \frac{1}{n-1}$

$+ \frac{1}{n-2} - \frac{1}{n}$

$+ \frac{1}{n-1} - \frac{1}{n+1}]$

$= \frac{1}{2} [1 + \frac{1}{2} - \frac{1}{n} - \frac{1}{n+1}]$

$= \frac{1}{2} (\frac{3}{2} - \frac{n+1+n}{n(n+1)})$

$= \frac{3}{4} - \frac{2n+1}{2n(n+1)}$

(biii)
As $n \to \infty$, $\frac{1}{n} \to 0$ and $\frac{1}{n+1} \to 0$, the sum of series tends to $\frac{3}{4}$, a constant. Thus, series is convergent.

(biv)

$\sum_{r=5}^{n+3} \frac{1}{(r-3)(r-1)}$

Replace $r$ by $r + 2$. Then we have

$\sum_{r=3}^{n+1} \frac{1}{(r-1)(r+1)}$

$= \sum_{r=2}^{n+1} \frac{1}{(r-1)(r+1)} - \frac{1}{(2-1)(2+1)}$

$= \frac{3}{4} - \frac{2(n+1)+1}{2(n+1)[(n+1)+1]} - \frac{1}{3}$

$= \frac{5}{12} - \frac{2n+3}{2(n+1)(n+2)}$

Its been awhile since we last posted. And it is good to know that JC1s have all been well inducted or settled into their schools. Of course, I do hear that many schools are severely overcrowded recently. Anyway, I thought of sharing how students can get ready for JC. I contemplated sharing how to cope with JC Mathematics, but decided to be more general this time round.

Firstly, JC life can be quite rigorous. With CCA and different subject commitments piling, students must try their best to stay healthy (get enough sleep) and juggle time (skip some dramas) efficiently. For science subjects (not just H2 Mathematics), students should avoid procrastinating. The schools do not go back to teaching the subjects again, maybe just refresh using questions or tests. Thus, seek help if you need and do not just sweep it off. For J1, your A’levels is pretty much in 22 months while for J2, it is 10 months. So the clock started ticking.

Secondly, last year’s papers were intuitive and some questions were driven to see if students do understand their content and can think on their feet. And we have a name for such questions, it is application questions. For H2 Mathematics, they have made an effort to allocate about 25% of the total marks to application questions. Thus, students need to shift their focus from doing to learning. It is important for them to appreciate the concepts in each topic.

Thirdly, I understand some students enter JC and realise that there are really some (or a lot of) smarter peers around. Do not feel pressured and just stay focus. Some of them might have found help, or developed better intuition for certain things. Comparing with your neighbour will only make yourself more stressed. This is unnecessary stress.

Lastly, JC is the last “school” you have. So do enjoy yourself. Pick a CCA that you really want to try. 🙂

Since Mr Teng teaches H2 Mathematics, here are some little tips for H2 Mathematics as I told my J1s this year.

1. Some topics from High School are still very relevant, which is why I gave a proper review test. These topics are considered under assumed knowledge for H2 Mathematics, and you can find them here. A good understanding of these topics will allow you to follow classes better. You will learn that schools are constantly rushing to clear topics.

2. Learn the topics. You do not need to master them, but learn and find out what is going on. Because you can memorise the entire Ten Years Series and realise that it will not save you.

3. You will learn that time is very precious during exams. In general, 1 mark is 1.5 mins. And you should not go beyond it for questions. Rather learn the hard way to time manage well during exams, start with your normal practices at home. Thats why I encourage my students in class to do fast. Your papers will be two 3-hour papers, so during that 3-hour, you must exhibit sufficient tenacity.

P.S. I’ve spent the last few months getting a lot of application questions up. Aside from sharing them with students in my classes, I’ll also put them here. So do check in. 🙂

Happy CNY!

### Call for Registration 2018!

Registration for classes in 2018 has been opened. You can find out more about the class schedules here. Do note that JC1 class will commence in first week of January as there are registrations from IP students. For O’levels students, you can treat it like a head-start! We will also be holding a workshop for Post-O’levels Release of Results. So do stay tuned!

Students/ Parents can call Mr. Teng at +65 9815 6827, or the centre at +65 6567 3606  for further enquiries!

Lessons will be held at:
Newton Apple Learning Hub
Blk 131, Jurong Gateway Road #03-263/265/267 Singapore 600131
Tel: +65 6567 3606

### Some TYS Questions worth looking at

Prelims Exams was scary. H2 Mathematics isn’t that easy.

Students that had difficulties finishing their prelims exams, should consider working on their time management. The best way to do it, practice 3 hour paper… in a single sitting. And students should note to modify their TYS slightly as several questions in each paper are out of syllabus. In general, we give ourselves 1.5min for every 1 mark.

So here, I’ll share a list of questions that Mr. Wee has compiled. Mr. Wee also wrote e-books recently on solving non-routine problems. They are very interesting and provides the learners a new perspective to solving problems.

Non-routine Problems (Click to link to the solutions)
N2016/P1/Q3
N2016/P1/Q8
N2016/P1/Q10(a)
N2015/P1/Q3
N2015/P1/Q11

Application Questions
N2016/P1/Q9
N2015/P1/Q8
N2014/P1/Q11
Specimen P1/Q9
Specimen P1/Q11
Specimen P2/Q9
Specimen P2/Q10

All the best for your revision!

### HCI GP Prelims paper 1 2017

1. ‘Nothing but provocation and self-centredness.’ Is this a fair description of the state of affairs in today’s world?
2. ‘My life, my choice.’ How far can people expect to live life this way?
3. Should historical monuments and objects be preserved when such an undertaking is very expensive or even a source of unhappiness?
4. ‘Many receive an education, but few are educated.’ Discuss with reference to situations in your society today.
5. ‘Tourism brings less developed countries more harm than good.’ Comment.
6. How worried should we be that recent advances in science and technology are creating new challenges and worsening old problems?
7. ‘Looks matter, and much more than substance too.’ Would you agree with this claim?
8. ‘The hallmark of a great country is not how prosperous it is, but how inclusive its people can be.’ Should your country work towards this ideal?
9. ‘We must surrender our human rights to win the battle against terrorism.’ Do you agree?
10. ‘Smart cities: innovative, but not necessarily better.’ What do you think?
11. ‘Corporate social responsibility is bad for business and companies should not be expected to take it up. To what extent would you agree?
12. ‘Let us read and let us sin, for what harm can these amusements bring?’ Comment.

### Anderson JC GP Prelims paper 1 2017

1. Should small countries be allowed to take the lead in global affairs?
2. To what extent can the Arts effect positive social change today?
3. ‘Experiences are more valuable than material possessions.’ Do you agree?
4. ‘People in the workplace should embrace, rather than fear, technological advancements.’ Discuss.
5. ‘The news today deals with what is popular, rather than what is important.’ How far do you agree with this statement?
6. Evaluate the claim that a more connected world has resulted in greater divisions.
7. ‘Public figures today are overly concerned about what people think of them.’ What is your view?
8. Consider the view that there is no value in slowing down in today’s competitive world.
9. Discuss the appeal and value of creativity in your society.
10. Considering the increasing threat of terrorism, are governments justified in limiting people’s rights?
11. To what extent is animal testing acceptable in scientific research?
12. ‘Economic development is favoured at the expense of the welfare of people.’ How true is this of your society?

### Temasek JC GP Prelim Paper 1 2017

Temasek Junior College 8807 H1 General Paper Paper 1 2017

1. Can government surveillance eradicate the threat of terrorism?
2. Examine the claim that globalization creates equal opportunities for all.
3. ‘The government is not doing enough to support local sportsmen in your society.’ What is your view?
4. To what extent is a universal language desirable?
5. Should people in your society be fearful of the future?
6. ‘Graciousness is lost as society progresses.’ Is this an accurate reflection of your society?
7. How far do you agree that technology gives us greater control in life?
8. Consider the view that what is posted online is all talk and no action.
9. ‘Failure should never be acceptable.’ Discuss.
10. Do you agree that only parents should be allowed to discipline their children?
11. Is volunteerism always good?
12. ‘The world today values appearance over substance.’ Is this a fair comment?

### Tampines GP Prelim Paper 1 2017

Tampines Junior College 8807 H1 General Paper Paper 1 2017

1. How realistic is it for your society to embrace diversity?
2. Protecting the environment should be given greater priority than eradicating poverty. How far do you agree?
3. ‘Appearance can be deceiving.’ To what extent is this true of the media today?
4. Discuss the view that smart devices have not made us smarter.
5. Education is the key to solving all social problems. Discuss.
6. Should firms have the responsibility to improve the quality of life of the communities they operate in?
7. Assess the view that literature is of little use to society.
8. Wealth is no guarantee of a better life. How far do you agree?
9. History is of little significance to a modern society. Discuss.
10. ‘Failure is always an option.’ Discuss.
11. A free and unrestricted media is essential for society to progress. How far do you agree?
12. How far is the arts a reflection of your society’s level of development?