**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 2014 to 2010
- S2 papers 2009 to 2007
- Mark scheme for June 2014 paper
- Mark scheme for June 2014 (R) paper

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

_{r=0}

^{n}r

^{2}.p(X=r)–mean

^{2}

Σ

_{r=0}

^{n}r

^{2}.p(X=r)

= Σ

_{r=0}

^{n}(r

^{2}–r).p(X=r)+ Σ

_{r=0}

^{n}r.p(X=r)

= Σ

_{r=0}

^{n}r(r–1).p(X=r)+ Σ

_{r=0}

^{n}r.p(X=r)

Do some algebra to calculate E(X

^{2}), and then calculate variance(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

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