Modal value & Expected value

Modal value & Expected value

JC Mathematics

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.

Scatter Diagrams

JC Mathematics

I was teaching scatter diagram to some of my students the other day. A few of them are a bit confused with correlation and causation. I gave them the typical ice cream and murder rates example, which I shared here when I discussed about the r-value.

Think of correlation like a trend, it simply can be upwards, downwards or no trend. And since we only discuss about LINEAR correlation here, strong and weak simply is with respect to how linear it is, that means how close your scatter points can be close to a line.

Since A’levels, do ask students to draw certain scatter during exams to illustrate correlation. Here is a handy guide.

Scatter Diagrams
Credits: pythagorasandthat.co.uk
Solutions to Set B

Solutions to Set B

JC Mathematics, Mathematics

Hopefully, you guys have started on the Set B. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. 🙂

If you do have any questions, please WhatsApp me. 🙂

Relevant Materials: MF26

Solutions to Set A

Solutions to Set A

JC Mathematics, Mathematics

Hopefully, you guys have started on the Set A. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. 🙂

If you do have any questions, please WhatsApp me. 🙂

Relevant Materials: MF26

Thoughts on the H2 Mathematics (9758) Papers 2017

Thoughts on the H2 Mathematics (9758) Papers 2017

JC Mathematics, Mathematics

Solutions can be found here.

Personal Thoughts: The paper isn’t tedious. Students can do them so long as they know their stuffs. There are several generalising of questions, like question 6 of paper 1. We also saw how conditional probability was actually tested subtly, this tests students’ abilities to reason with guidance (not sure if after this first trial year, will they still guide the students.) Application questions were not tough and well guided. Students can solve it easily if they read it well. Statistics was well crafted and neat.

To be blunt, I’ll give credit to the 9740 H2 Mathematics paper that run concurrently, since it is too tough to set two sets of papers. Its easy to acknowledge that the 9740 (2016) paper was way harder than 9740 (2017). Next year won’t be the same.

Advice: Students should be careful when you revise, make sure you learn, and not do. Understand what you’re doing. The 2017 paper was an inquisitive paper, examiners were watching closely if you pay attention to details, and know your definitions well.

I’ll do an analysis for the paper, you can click on the individual question and read. For students that took the paper, I hope it doesn’t demoralise you.

Paper 1

Paper 2

 

 

2017 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

2017 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

JC Mathematics, Mathematics

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

Numerical Answers (click the questions for workings/explanation)

Question 1:
Question 2:
Question 3:
Question 4:
Question 5:
Question 6: \mu = 1.69, \sigma^2 = 0.0121
Question 7: 0.254; 0.194; 0.908
Question 8: 40320; 0.0142; \frac{1}{4}
Question 9: \text{r}=0.978; a=0.182, b=2.56; $293
Question 10: 0.0336; \bar{y}=0.64, s^2 = 0.0400; Sufficient evidence.
Question 11: \frac{48+x}{80+x}, \frac{32+x}{80+x}; x= 16; \frac{25}{32}; \frac{7}{16}; \frac{341}{8930}
Question 12: 0.773; 0.0514; 0.866; 0.362

 

Relevant materials

MF26

KS Comments

Thoughts on H2 Mathematics (9758) 2017 Paper 1

Thoughts on H2 Mathematics (9758) 2017 Paper 1

JC Mathematics, Mathematics

This is a new syllabus and this is the first time it will be tested. Personally, I don’t think it will be easy and students should not underestimate this upcoming A’levels. And I’m referring to the A’levels, on the whole. We saw how the Science Paper 4 were… unexpected.

The new H2 Mathematics (9758) syllabus has several topics removed, and these were mostly topics that were “drill-able”, aside from complex numbers. The new syllabus added in mainly, new integration forms, focus on parametric Equations with Cartesian equations, and of course, Discrete R.V. But let us leave the statistics out.

Students should familiarise themselves with the trigonometry Formulae in MF26. There are several topics that can be linked up with trigonometry, makes me wonder why it isn’t a chapter by itself. Complex numbers has a trigonometry form too, so make sure students know how to manipulate it, given the trigonometry Formulae.

Next, students should understand the use of Maclaurin’s. What does it mean for x to be small, and the implications when they say h is small compared to R… And also finding the general term of a Maclaurin’s Expansion.

It won’t hurt to review how to find the Area using Shoe-lace method. And not forgetting our Sine Rule and Cosine Rule.

Do know how to prove a one-one function… Non-graphically. (i.e. not using the Horizontal Line Test)

Do know that the oblique asymptote of f(x) becomes y=0 when we do the y = \frac{1}{f(x)} transformation too.

Lastly, students must READ really carefully and discern every information. Having marked many scripts, many students do not read carefully and lose marks here and there. And they do add up… Be alert and read, take note of the forms that they want. Here are 10 little things to take note when you read the question.

  1. Cartesian/ Polar/ Exponential for complex
  2. Scalar/ Parametric/ Cartesian for vectors
  3. Set/ range/ interval of values
  4. Algebraically => show all the workings without a GC.. usually discriminant, completing the square or maybe some differentiation will be involved.
  5. Without using a calculator => show your workings and check with a GC (secretly)
  6. Decimal places, etc…
  7. Rounding off when you’re dealing with an inequality
  8. Units used in the questions, (ten thousands, etc)
  9. Rate of change; leaking means the rate is negative…
  10. All answers should be in 3 SF UNLESS OTHERWISE STATED. Degrees to 1 DP. RADIANS to 3 SF.

Have fun and all the best!