### 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.

### Scatter Diagrams

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.

### A-level H2 Mathematics (9758) Suggested Solutions (2017)

Here is the suggested solutions for H2 Mathematics (9758). They are all typed in LaTeX, so if it does not render, please leave a comment and let me know. Thank you.

The suggested solutions for H2 Mathematics (9740) is here.

Students of mine should obtain the modified A’levels Paper, and the solutions to the additional questions can be found here.

Year 2017

MF26

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

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: $0.773; 0.0514; 0.866; 0.362$

MF26

### 2017 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions

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: $2 \sqrt{15}; xy=6$
Question 2: $d = 1.5;~ r \approx 1.21 \text{~or~} r \approx -1.45;~n=42$
Question 3: $(\frac{1}{2}a, 0), (0,b);~ (a+1, 0,);~ (\frac{a+1}{2}, 0);~ (0, a), (b, 0);~ a = 1;~ gg(x) = x, x \in \mathbb{R}, x \neq 1 , ~ g^{-1}(x) = 1 - \frac{1}{1-x}, x \in \mathbb{R}, x \neq 1;~b= 2 \text{~or~}0$
Question 4: $15.1875;~ \frac{\pi}{2a(a-1)};~ b = \frac{1}{2} + \frac{1}{2}\sqrt{1-a+a^2}$
Question 5: $0.647;~ 0.349;~k=2.56$
Question 6: $955514880;~ 1567641600;~ \frac{1001}{3876}$
Question 7: $31.8075, 0.245;~ p = 0.0139$; Do not reject $h_0$, Not necessary.
Question 8: Model (D); $a \approx 4.18, b \approx 74.0;~ r \approx 0.981$
Question 9: $0.632;~ 1.04 \times 10^{-4};~ 0.472;~ 0.421;~ 0.9408$
Question 10: $0.345;~ 0.612;~ \mu = 12.3, \sigma = 0.475;~ k \approx 55.7$

MF26

### Solutions to the modified A’levels Questions

Students of mine who have been diligently doing the modified TYS I sent them, and have difficulties with the questions that were added in to make the paper a full 3 hour paper, will find the following solutions helpful. Please try to do them in a single 3 hour seating, these are modified to cater to the 9758 syllabus…

The rest of the solutions (that are questions from the original TYS) can be found here.

2012/P1/Q10

2012/P2/Q2

20112/P2/Q7

2012/P2/Q7

### 2017 A-level H2 Mathematics (9758) Suggested Solutions

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

Paper 1

Paper 2

MF26

### 2017 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

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: $2 \sqrt{15}; xy=6$
Question 2: $d = 1.5;~ r \approx 1.21 \text{~or~} r \approx -1.45;~n=42$
Question 3: $(\frac{1}{2}a, 0), (0,b);~ (a+1, 0,);~ (\frac{a+1}{2}, 0);~ (0, a), (b, 0);~ a = 1;~ gg(x) = x, x \in \mathbb{R}, x \neq 1 , ~ g^{-1}(x) = 1 - \frac{1}{1-x}, x \in \mathbb{R}, x \neq 1;~b= 2 \text{~or~}0$
Question 4: $15.1875;~ \frac{\pi}{2a(a-1)};~ b = \frac{1}{2} + \frac{1}{2}\sqrt{1-a+a^2}$
Question 5: $\frac{5}{12}, \frac{5}{14}, \frac{5}{28}, \frac{1}{21};~ \mathbb{E}(T) = \frac{20}{7}, \text{Var}(T) = \frac{75}{98};~ 0.238$
Question 6: $955514880;~ 1567641600;~ \frac{1001}{3876}$
Question 7: $31.8075, 0.245;~ p = 0.0139$; Do not reject $h_0$, Not necessary.
Question 8: Model (D); $a \approx 4.18, b \approx 74.0;~ r \approx 0.981$
Question 9: $0.632;~ 1.04 \times 10^{-4};~ 0.458;~ 0.421;~ 0.9408$
Question 10: $0.345;~ 0.612;~ \mu = 12.3, \sigma = 0.475;~ k \approx 55.7$

MF26