## Assumed knowledge

- An informal understanding of probability, as covered by the series of TIMES modules Chance (Years 1–10).
- Some familiarity with sets, set notation and basic operations on sets, as covered by the TIMES module Sets and Venn diagrams (Years 7–8).

## Motivation

### Chance in everyday life

Ideas of chance are pervasive in everyday life, and the use of chance and risk models makes an important impact on many human activities and concerns.

- When tossing a fair coin, what is the chance that the coin lands heads?
- What is the probability that I win the lottery?
- When playing Tetris, what are the chances of three long sticks (piece \(\mathtt I\)) in a row?
- What chance do I have of getting into my preferred tertiary course?
- What chance do I have of getting into my preferred tertiary course if I get a score of at least 87 at the end of Year 12?
- What is the probability that the Reserve Bank will cut interest rates by half a percent?
- What are the chances of getting a sexually transmitted disease from unprotected sex?
- If I leave my school bag at the bus stop briefly to run home and get something I've forgotten, what are the chances of it still being there when I get back?
- If I leave my Crumpler bag at the bus stop briefly to run home and get something I've forgotten, what are the chances of it still being there when I get back?
- What is the probability of my bus being more than 15 minutes late today?
- What is the chance that my football team will win the premiership this year?
- What is the chance that Tom/Laura will like me?

There are important distinctions between some of these questions. As we shall see in this module, there are several different approaches to probability.

In this module we primarily deal with **idealised models** of probability based on concepts of symmetry and random mixing. These are simplified representations of concrete, physical reality. When we use a coin toss as an example, we are assuming that the coin is perfectly symmetric and the toss is sufficiently chaotic to guarantee equal probabilities for both outcomes (head or tail). From time to time in the module, we will hint at how such idealised models are applied in scientific enquiry. But that is really a topic for later discussion, in the modules on sampling and inference (Random sampling, Inference for proportions, Inference for means).

A second way of obtaining probabilities is as an extrapolation from a relative frequency. This really involves inference, but it is one way that probability and chance ideas are used in everyday life. For example, when the Bureau of Meteorology predicts that the chance of rain tomorrow is 20%, there is no clear, simple procedure involving random mixing as in the coin toss. Rather, there is a long history of conditions like those leading up to tomorrow. So the reasoning is, approximately: 'Among all of the previous instances of weather conditions similar to those we are experiencing now, there was some rain on the following day in 20% of instances.' Hence, by a form of inductive reasoning, we say that the probability of rain tomorrow is 20%.

These first two approaches can converge in some cases. Even if we are unconvinced by propositions about the randomness of the coin toss and the symmetry of the coin, we may have observed that the proportion of heads obtained from many similar tosses is close to 0.5, and therefore assert that the probability of obtaining a head is one half.

A third method of obtaining probabilities is subjective. This use is generally more casual. I may say that there is a 90% chance that Laura will like me, but you may disagree. Betting sometimes involves subjective probabilities on the part of the person making the bet. We do not consider subjective probability further in this module.

### Chance in the curriculum

Ideas about chance or probability receive extensive coverage in all of the previous years of the Australian Curriculum. In this module we develop these ideas more formally; this is the stage in the curriculum where a structured framework for probability is presented.

Even in the very early years of the curriculum, there is coverage of basic ideas such as the words we use to describe degrees of certainty — words such as 'might', 'is likely' and 'definitely'. Students learn how to associate these words with events in their own experience, such as 'Our teacher **might** be sick today' or 'I will **definitely** sleep tonight'.

As these ideas are developed further, students will gain an understanding of chance or probability in relation to events: phenomena that we can observe. They learn that probability is measured on a scale from zero to one, and that impossible events have a probability of zero, while events that are certain have a probability of one.

The treatment in this module follows the path of conventional probability theory. The standard axioms are introduced, the important properties of probability are derived and illustrated, and the key ideas of independence, mutually exclusive events and conditional probabilities are covered. These are all foundational ideas for the other modules on probability and statistics.