You may find these two summary sheets useful:

“Do and don’t” summary of the course

and these notes

How many marks does Edexcel take off for bad writing in S2?

**Week 1** (22-23 November)

What S2 is about. Four questions which we’ll be able to answer once we’ve learned the concepts.

Practical: biased dice.

What is a random variable? It’s really a function rather than a variable. It describes possible outcomes from an activity and attaches probabilities to them so that the probabilities add up to 1.

Binomial distribution: what it is

When to use it

How to calculate by hand

How to use the Classwiz calculator for it

Ex.1B

Mean and variance

Ex.1E

Answer the first of our “four questions”.

Poisson distribution: what it is

When to use it

How to calculate by hand

How to use the Classwiz calculator for it

Ex.2C

Mean and variance

Ex.2D, 2E, 2F

Answer the second of our “four questions”.

Click here for example of use of Poisson and binomial distributions

Normal distribution: what it is

When to use it

How to use the Classwiz calculator for it

Relate to binomial and Poisson (also mention Central Limit Theorem, not on syllabus). The rules of thumb for approximations are not stated in the S2 book, and you’re not expected to know them, but they’re usually taken as:

- Poisson approx for binomial – n big and p small, meaning roughly n ≥ 20 and p ≤ 0.05, and np < 10
- Normal approx for binomial – n big and p not too small, meaning roughly np > 5 and nq > 5
- Normal approx to Poisson – λ large, meaning roughly λ > 10

Continuity corrections: if you use a continuous distribution like the normal distribution as an approximation to calculate probabilities for a discrete variable (like binomial or Poisson), then you should calculate normal-distribution probability for the range of continuous values which would *round to* the discrete values you’re inquiring about. To find P(Discrete Variable=12), take P(11.5 < Normal Variable < 12.5). To find P(Discrete Variable between 12 and 15), take P(11.5 < Normal Variable < 15.5). To find P(Discrete Variable less than 15), take P(−∞ < Normal Variable < 15.5).

Ex.5C, 5D

Answer the third of our "four questions"

Continuous uniform distribution: what it is

When to use it

How to calculate it

Mean and variance

Recap and intro to CUD (pdf)

Ditto (odt)

and answers to ditto

Continuous Uniform Distribution questions

Which distribution? My own worksheets (below) and Ex.4C Q.5, 6, 9

Sampling: terminology (which is really all chapter 6 is)

Hypothesis testing: what is a statistic? H_{0}, H_{1}, p, one-tailed, two-tailed, critical value, critical region

Answer the fourth of our “four” questions

Choc chip cookie investigation and debrief (pdf)

Ex.7D

Mini-test on ground covered so far.

Explain why simple p-value hypothesis testing is unsatisfactory (not in syllabus)

Continuous random variables

Pdf, cdf, mode, quartiles, E(aX+b), Var(aX+b)

Ex.3E

Mop-up, practice, test

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**Collections of past-paper questions by topic**

- Sampling and hypothesis testing
- Continuous random variables
- Distributions: binomial, Poisson, normal, uniform

**Collections of full past papers by date**

- S2 papers 2017 to 2010, with mark scheme for 2017 paper but not others
- Mark scheme for June 2014 paper
- Mark scheme for June 2014 (R) paper
- S2 papers 2014 to 2010
- S2 papers 2009 to 2007

Mark schemes for other S2 past papers can be found at the Physics and Maths tutor website

**Week 1**: How many throws do you need to be confident that a biased dice is biased?

Or: how weird does the result have to be to make you confident that the result is not random?

What does “random” mean for a fair dice?

The binomial distribution; coin tosses and dice throws.

- Pascal’s triangle
- Calculating probabilities using nCr
- Calculating probabilities using n!/(n-r)!r!
- Calculating probabilities using tables

**When to use the binomial distribution**

Starter: The binomial distribution describes numbers of heads (or tails) when you toss a coin, or the number of ones (or sixes, or whatever) when you toss a dice.

Write on the whiteboard some other random variables described by a binomial distribution

Write on the whiteboard what the rules are for a random variable to be described by the binomial distribution.

When we’re sure we have it correct and clear, you will write those rules in your notebook.

**Review first problem on the “four problems” sheet**

**Mean and variance of the binomial distribution**

Mean = np and variance = np(1−p)

Standard deviation = √variance = √[np(1−p)]

For the exam, you need only read these formulas from the formula book. But this is how to get them.

If X is a random variable described by the binomial distribution for n trials and probability of success p, which we call **B(n,p)**:

p(X=r) = nCr.p^{r}(1–p)^{n-r} = [^{n!}/_{(n–r)!r!}]. p^{r}(1–p)^{n-r}

**The mean of X = E(X)** = Σ_{r=0}^{n} r.p(X=r)

Do some algebra to calculate E(X)

**The variance of X = E(X ^{2})–[E(X)]^{2}**

= Σ

Σ

= Σ

= Σ

Do some algebra to calculate E(X

**Homework:**

S2: Binomial distribution

Ex.1B Q.1-3

Ex.1C Q.1-3

Ex.1D Q.1-3

Ex.1E, Q.4, Q.7

Optional extra: Review exercises, p.64 Q.4, Q.8

**Week 2**

**Get good with the binomial distribution**

**Starter activity**: Shut your notebook and textbook and write on the whiteboard:

1. when a random variable follows the binomial distribution

2. what B(n,p) means

3. what the formula is for the probability of r successes in n trials when the probability of success in each trial is p

4. what the mean of B(n,p) is

5. what the variance of B(n,p) is

6. what equation connects standard deviation and variance.

**Activity**

Ex.1E Q.8,9. Review exercises p.64 Q.1, Q.2 without using an approximation.

**Starter**

We want to find the probability of a team scoring 3 goals in a football match of 90 minutes. As an approximation, divide the 90 minutes into 30 units of 3 minutes each and take the probability of a goal in each 3-minute unit to be 1/30.

We will then work through an introduction to, and activities about, the Poisson distribution.

**Homework for next week**: Binomial and Poisson distributions. Review exercises p.64 Q.1,2 , Ex.1E Q.9, Ex.2E Q.1-4.

**Week 3**

**Getting good with the Poisson distribution, and Poisson approximation to binomial**

**Starter**: Shut your textbook and your notebook, and write on the whiteboard what variables other than goals in football matches might follow a Poisson distribution

We will do Ex.2F Q.2 as a worked example in class.

Then work on Ex.2E Q.1-5.

Then we will summarise the Poisson distribution:

- A random variable X follows a Poisson distribution if it is a count of the number of successes where:
- Successes occur randomly in continuous time or space (i.e. not in a discrete series of trials)
- Successes are independent of each other, and occur at a constant rate over time or space

- Formula: probability of r successes = [λ
^{r}/r!].e^{−λ} - Mean=λ. Variance=λ. Standard deviation=√λ
- Poisson can be used as a simple approximation to binomial when n is big and p is small. There is no exact rule for what “big” and “small” mean here, and Edexcel does not demand one. But a good rule of thumb is n≥20 and p≤0.05.

**Getting good with the Poisson distribution, and Poisson approximation to binomial**

**Starter**: Shut your textbook and your notebook, and write on the whiteboard the conditions for a random variable to follow the binomial distribution, and the conditions for it to follow the Poisson distribution.

Then we will work on Ex.2F Q.1-10.

**Week 4**

The normal distribution, and normal approximation to binomial

The normal distribution is defined by the probability distribution function

The indefinite integral of the function cannot be expressed in terms of functions we already know, but we can calculate the definite integral

and thus see where the factor comes from. It’s a neat example of the occasional pattern in maths where we can solve a problem, paradoxically, by translating into a more complicated problem.

Given that definite integral, a bit of integration by parts will confirm that the variance = 1.

Go over normal distribution from S1 again

Visual: binomial approximation to normal

Central Limit Theorem: why the normal distribution is so widely used, and the pitfalls

Continuity correction

Homework: Ex. 5B Q.1-4

**Week 5**

- Discrete and continuous, and which distribution approximates which?
- Which distribution when? Binomial, Poisson, and normal

Discrete and continuous, and which distribution approximates which?

Problem 3 on our “Four Problems” sheet

Continuous uniform distribution

Homework: Continuous uniform distribution: Ex.4C Q.1, 2, 3

**Week 6**: Words to use about sampling. Homework: Review Exercise, p.127, Q.6, and p.128 Q.14, page 129 Q.17. Exam practice paper p.131, Q.1

**Week 7**: no lessons (half-term)

**Week 8**: Basics of hypothesis testing. Problem 4 on our “Four Problems” sheet. Null hypothesis, alternative hypothesis, significance level. Homework: Ex.7A, Q.1-7.

**Week 9**: More on hypothesis testing. Critical values, critical region, actual significance level. Ex.7B Q.1-3, 8-10, 11-12.

**Week 10**: Continuous random variables. Deriving cdf F(X) from pdf f(x). Deriving parameters of pdf from F(∞). Deriving E(X) as integral from −∞ to ∞ of xf(x)dx. Deriving E(X^{2}) as integral from

−∞ to ∞ of x^{2}f(x)dx. Deriving var(X) as E(X^{2})−(E(X))^{2}. Deriving median of X from F(m)=0.5, and lower quartile and upper quartile similarly. Deriving mode of X from maximum of f(x). Recalling definition of negative and positive skew from S1. Homework: continuous random variables. Ex.7D Q.1 and 2, plus four questions on hypothesis testing from Ex.3E in addition to what’s been done in class.

**Week 11**: no lessons (mock exam week). Homework: work through mock exam paper and mark scheme.

**Week 12**: “How random can you get?” survey. Review mock exam. Homework: an S2 paper.

**Week 13**: “How random can you get?” survey. Homework: an S2 paper

**Week 14**: Another mock exam paper. Revision sheets: Binomial, Poisson, normal: what you have to remember; and Revision sheet for the whole course. S2 paper in class, and S2 test paper in exam conditions.

**Week 15**: no lessons (end of term on Thursday)

- Four problems which define what you will learn on the course
- Introduction to, and activities about, the Poisson distribution
- Worksheet on discrete and continuous, and Poisson and normal approximations to binomial. This uses these websites http://www.waldomaths.com/PoissBin1L.jsp: binomial approximation to Poisson – visual demonstration; http://www.waldomaths.com/Binomial1L.jsp: binomial approximation to normal – visual demonstration.
- Central Limit Theorem – visual demonstration
- The Central Limit Theorem and the normal distribution
- Exam questions on the continuous uniform distribution
- Introduction to Benford’s Law
- The maths of Benford’s Law
- “How random can you get?” survey
- Do costlier choc chip cookies have more choc chips? Debrief
- How not to do hypothesis testing: the “21 grams” experiment

**Back-up materials**

- Website to help you learn the definitions you need for the exam
- Exam questions on the binomial distribution
- Proof of E and var for Poisson distribution, and that binomial probabilities approach Poisson probabilities as n gets larger with np fixed.
- Analysis of Premier League goals 2013-4
- Other hypothesis testing projects
- Binomial, Poisson, normal: what you have to remember
- Revision sheet for the whole course

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and

this test paper on decision maths.

Please mark your own work before handing it in. Answer sheets here:

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Sampling and hypothesis testing: Exercise 7D Q.1-10 and Exercise 6C Q.6, 7, 8.

Exercise 7D Q.1-10 and Exercise 6C Q.6, 7, 8

And the “Examination practice paper” at the end of the S2 book, p.131-132, questions 1 to 6.

You may find these two summary sheets useful:

“Do and don’t” for S2 statistics

and these notes

How many marks does Edexcel take off for bad writing in S2?

Then please do these more demanding tasks: two FP2 practice papers compiled from Review Exercise questions in the book.

I’m proposing these because answers can be found in the back of the textbook, and further help on Solution Bank.

And the June 2016 FP2 paper (which was a harder-than-usual paper). You can find the paper, plus worked answers and a mark scheme, at:

https://mathsmartinthomas.wordpress.com/2016/12/07/june-2016-fp2-notes/

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FP2 June 2017: click here for pdf

FP2 June 2017 worked answers: click here for pdf

FP2 June 2017 mark scheme: click here for pdf

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Two trains are 100 miles apart. They travel towards each other on the same track, each at 20 miles per hour. A fly starts on the left-hand train, buzzes to the right-hand train, turns around immediately, flies back to the first train, and goes on flying back and forth between the two trains until they collide. If the fly’s speed is 40 miles per hour, how far will it travel?

Prizes were won by Mohaned al-Bassam, Genie Louis de Canonville, Bolaji Atanda, Tegan Hill, Reece Jackson, Howard Tran (and one other student whose name I’m temporarily forgotten).

The time until collision is 100/40, since the relative speed of the trains is 40 mph.

The fly always travels at 40mph, so in a timespan of 100/40 it will travel 100 miles.

This can be worked out from an infinite series: time of fly’s first flight from one train to another plus time of fly’s second flight between trains plus time of fly’s third flight between trains…

Bolaji did it that way. The story is that when the mathematician John von Neumann was first asked this puzzle, he gave the answer instantly. The questioner said: Aha, I see you found the quick method, rather than defining and summing the infinite series. John von Neumann said: What quick method? I just summed the infinite series in my head.

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**Year 12 Further Maths homework to Wednesday 6 December**

Part B: Decision maths – complete Ex.1D Q.1-4

Part A: Little individual bits of Part A homework are written in the homework feedback in the back of your books. If you see no request there to do a further problem, then no Part A for you.

Part C: New worksheet of induction questions from past papers

Be sure to write each proof clearly:

Claim

Step 1: Prove – true for n=1

……… □

Step 2: Prove – true for n=k ⇒ true for n=k+1

To prove: [write out the claim for n=k=1]

True for n=k ⇒ ….

⇒ ….

⇒ ….

⇒ …. □

The claim is true for n = 1.

For all k, we have shown that if it is true for n = k then it is true for n = k + 1

So, by mathematical induction, it is true for all n ∈ ℕ

**Year 13 Further Maths homework to Wednesday 13 December**

Part C: FP2 June 2015 paper, here: https://mathsmartinthomas.wordpress.com/2017/11/27/fp2-june-2015-paper/. Please do this on Thursday 30 November, at the time when we’d usually have a lesson. The mark scheme will be available on this website from 15:10 on Thu 30 Nov. Please mark your own work.

And FP2 revision exercises, here: https://mathsmartinthomas.wordpress.com/2017/11/21/revision-homework-for-y13-further-maths/

Part A: see https://mathsmartinthomas.wordpress.com/2017/11/29/part-a-homework-for-y13-further-maths-for-6-dec-2017/ for individual Part A homework for each student.

Part B: S2 Ex. 5B Q.1-2, 5C Q.1-3, 5D all. (You can do this after your Further Maths exam on Tuesday, to be complete by Wed 13 Dec, if you like).

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Bolaji – try FP2 Review Exercises 2 Q.60c, d, e again using my comments. Do FP2 Ex.6F Q.8 and Review Exercises 2 Q.51

Michelle – Try FP2 Review Exercises Q.61 again, and Tegan – try FP2 Review Exercises 2 Q.61c again. Do the FP2 revision questions at https://mathsmartinthomas.wordpress.com/2017/11/21/revision-homework-for-y13-further-maths/.

Onesimus – Try FP2 Ex.6F Q.8 again using my comments.

Umut – try FP2 Review Exercises Q.60e again using my comments. Do the FP2 revision questions at https://mathsmartinthomas.wordpress.com/2017/11/21/revision-homework-for-y13-further-maths/. Do S2 Ex.2C Q.1-3, Ex.2D Q.1-3, Ex.2E Q.1-3.

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Revision homework for Y13 Further Maths in preparation for FP2 test on Tuesday 5 December 2017

This revision homework is to be done in the week starting Wednesday 22 November and the week starting Wednesday 29 November. It is additional to the (light) current S2 homework you will have in those weeks.

As always, email me if you need help.

Please mark your work against the answers in the back of the FP2 book, and the FP2 Solution Bank (available online). Please also let me see your work. Hand in on Wednesday 29 November what you have done by then, and scan and email the rest to me by Sunday evening 3 December so I can give you feedback before the test.

The selection of questions is deliberately focused on plain-vanilla methods, with nothing on the fancy bits (shortcuts for calculating Maclaurin series, ranges of validity for Maclaurin series, angles in circles and complex loci, that sort of thing). It also omits questions on the basic De Moivre’s Theorem calculations – questions like finding cos 5θ as an expression in powers of cos θ, or cos^{5}θ as an expression with cos 5θ, cos 3θ, cos θ terms. I think you’re all pretty good with those.

You’re welcome to do more Review Exercises for practice, but I’d strongly advise *not* doing that until after you’ve nailed down the basic plain-vanilla stuff as in the examples below.

**Roots of complex numbers 1**

Review Exercises 1 Q.33

**Möbius transformations 1**

Review Exercises 1 Q.34. (You can do this one in your head using our geometric method, but you’d better practise doing it the long Edexcel way).

**Maclaurin series 1**

Ex.6F Q.8. (This is a plain-vanilla differentiate-and-differentiate again question. You should find it easier after having done C3 differentiation).

**Polar coordinates 1**

Review Exercises 2 Q.51

**Roots of complex numbers 2**

Review Exercises 1 Q.35

**Möbius transformations 2**

Review Exercises 1 Q.45. (Sneakily, they give you the transformation z ↦ w, and then ask you about the transformation w ↦ z. Again, do it the long Edexcel way, though you can use our quick geometric method to check. The z diameter ends for part b are −2 and −i).

**Maclaurin series 2**

Ex.6F Q.14. (Again, plain-vanilla differentiate-and-differentiate again).

**Polar coordinates 2**

Review Exercises 2 Q.52

**Roots of complex numbers 3**

Review Exercises 1 Q.41

**Möbius transformations 3**

Review Exercises 1 Q.46. (Remember, anything like arg(z−a)=b is a half-line. Everything in part b depends on a good diagram. Again, show working for doing part c the long Edexcel way, though with our quick method you should be able to calculate in your head that the answer is u = 1/3).

**Maclaurin series 3**

Review Exercises 2 Q.47. (The method here is again plain-vanilla differentiate-and-differentiate again; it’s just more complicated differentiation than you’ve been used to).

**Polar coordinates 3**

Review Exercises 2 Q.55

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**Year 12 Further Maths homework to Wednesday 22 November**

Parts A, B, and C

**Part A**

Abass: Re-work Further Vieta Formulas (making new equations) exercise 4a(iii) by putting y=x+2, so x=y-2, and creating an equation for y.

Aniqa: no part A.

Enoch: Re-work Further Vieta Formulas (making new equations) exercise 4a(iii) by putting y=x+2, so x=y-2, and creating an equation for y.

Callum: Prove by induction that:

Helen: Do Further Vieta Formulas (making new equations) exercise 4a (ii) and (iii). Re-write, in your book, the proofs by induction that

Prove by induction that:

Howard: no part A

Iris: no part A

Javonne: no part A

Jeffrey: finish #51-60 of the exercises in the induction workbook on factorisation and index laws.

Lara: finish #51-61 of the exercises in the induction workbook on factorisation and index laws. Re-work Further Vieta Formulas (making new equations) exercise 4a(iii) by putting y=x+2 and creating an equation for y.

Mya: Re-work Further Vieta Formulas (making new equations) exercise 4a(iii) by putting y=x+2, so x=y-2, and creating an equation for y.

Obi: Re-work Further Vieta Formulas (making new equations) exercise 4a(iii) by putting y=x+2, so x=y-2, and creating an equation for y

**Part B**: Complete proof-by-induction workbook up to but not including the last page. (In class, I said: including the top half of the last page, but that’s too much. Just up to but not including the last page).

**Part C**: Q. 4b parts (i), (ii), (iii) from “Further Vieta Formula Exercises” worksheet https://mathsmartinthomas.files.wordpress.com/2014/09/vieta-formula-exercises.pdf (answers at https://mathsmartinthomas.wordpress.com/homework-2/#vfa. If you’ve already done this, good for you – no Part C for you).

Polar coordinates: Ex.7F Q.4-6, Review Exercises Q.60, 61.

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