### Introduction

This STEP support module covers arrangements and some problems with probability. Previous assignments can be found here, but you can do this one without having done the others first.

STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").

### About this assignment

The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.

The **warm up** for this assignment practices algebraic simplification and simultaneous equation solving.

The **main STEP question (2005 STEP 1 Question 1)** is about arrangements, and carefully considering all the possible options. There are some examples on the last page for those of you who have not met these ideas yet.

The **final question** is about conditional probability, and how very different $P(A|B)$ and $P(B|A)$ can be.

For more on some of the implications on confusing $P(A|B)$ and $P(B|A)$, see Prosecutor's fallacy , or for a shorter explanation see this page and this article . In particular, the probability that a defendant is innocent given that there is a DNA match can be very different to the probability that we get a DNA match given that the defendant is innocent - however it is the first one we want to consider! A discussion by David Spiegelhalter on why breast screening might do more harm than good can be found here.

An article about Bayes's Theorem and the implications for the results of Covid-19 testing can be found here.

People can be very bad at understanding probabilities and risks, visualising probabilities can make them much easier to understand.

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 6”** file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.

Here is a Worked Video Solution to the STEP question from this assignment.