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# Understanding Probability and Randomness

## 15-19 February 2021

This February half term we will be exploring the exciting subjects of Probability, Statistics and Randomness. In both programming and maths we will look at why these topics are so important, how they work and where they appear in the world around us.

As ever, participants will be divided into small classes of six or fewer, based on their age and experience in maths and programming.

Here’s what children joining this camp will be learning:

Mathematics

We will look at Probability and Statistics as well as Randomness and ask:

• is it possible to make an accurate guess?
• can we describe the world if instead of deterministic rules it is governed by random processes?

In all year groups we will be running two separate maths classes each day to study:

1. The Quest of Randomness, and
2. Problem Solving

Working on both Randomness and Problem Solving will provide a strong education and these two fields will strengthen each other.

Here’s what we’re aiming to do in each age group:

Year 3-4

Younger pupils will work on maths questions in which uncertainty is involved. The goal will be to extract useful information from a problem even when the data is incomplete. We will guess a number, use the average, apply the pigeonhole principle and play games so that pupils can appreciate the power of taking a mathematical approach to manage uncertainty.

Year 5-6

The Quest of Randomness: pupils will make guesses in advance about random events then perform experiments with the group collecting data to see the pattern of random results. Pupils will frame a mathematical model suitable for the pattern and then play games to find the best strategies and the chances that need to be considered. Pupils will need 5 coins, 5 dice, paper and a pen during this course.

Problem Solving: Combinatorics - can you find all the possibilities? Can you find all the permutations?

Year 7-9

The Quest of Randomness: starting from the simplest experiments pupils will arrive at complicated random events. On the way they will reveal the mathematical structure that lies behind the scenes and learn how to make professional guesses. This will include the beginnings of Pascal’s triangle and infinity.

Problem Solving: Sets, Combinatorics and Probability. Identities of Boolean Algebra.

Year 10-11

The Quest of Randomness: pupils will investigate different distributions and the possible shapes of randomness. Then they will consider processes that are governed by chance. How can we predict the result of a game if each step of it is uncertain?

Problem solving: Simultaneous systems of equations, Combinatorics and Probability.

Programming

Have you ever wondered what the random() function does in programming and how it works? What is a random number and how is it obtained? Children joining this camp will be invited to imagine a future job interview question such as ‘write a random number generator in a couple of lines of code’. Is it really that difficult to do this? And what is randomness? Is it actually possible to predict it? In this exercise, pupils learn how to answer these questions.

We will start with a basic programming structure that generates the random binary numbers 0 and 1. This sequence is very predictable but once we have given pupils a basic programme to do this, they will learn how to modify it to make the sequence more unpredictable. To make it work, pupils will define a custom function and use loops and variables.

Finally, the Linear Congruential Generator algorithm will be introduced to pupils. They will learn the working principle of the generator and will be able to use this knowledge to implement the random number generator in their projects.

The Random Walk concept is a very powerful maths object which can be used in many fields, for example:

• modelling the trajectory of a molecule in a liquid or gas (Brownian motion)

• finding the paths animals take during foraging

• fluctuations in stock prices in the stock market

All these cases can be approximated by random walk models, even though they may not be completely random in real life.

In this exercise, pupils learn the basic principle of the Random Walk algorithm. Children will have an example application and several tasks to solve. To solve the tasks, pupils will need to decompose the task itself from a given video and apply the Random Walk Algorithm. Pupils will even learn how to define the moving space, each step and the length of each movement.

The format for senior online camp is as follows:

• each day runs for six hours with four one hour classes and strategic breaks from the screen between them

• class sizes are six pupils or fewer of similar age and ability

• all classes are interactive, using Zoom, with pupils solving problems with the teacher and with each other

• class times are: 10.00-11.00, 11.30-12.30, 13.30-14.30, 15.00-16.00

• Fees are £300 for three days, £400 for four or £500 for five