## 16.1 Introduction

In this chapter,
*probability* is discussed.
In short,
*probability*
quantifies the chance that something
(an ‘event’)
will happen in the future.

More formally,
probability is discussed in the context of a procedure
whose result is unknown beforehand.
A list of all possible results from this procedure is called the
*sample space*.
An *event*
is then defined as any combination of these elements of the sample space.

**Definition 16.1 (Sample space)**The

*sample space*is a list of all possible and distinct results after administering a procedure, whose result is unknown beforehand.

**Definition 16.2 (Event)**An

*event*is any combination of the elements in the sample space.

**Example 16.1 (Sample spaces and events) **Consider rolling a fair, six-sided die.
We do not know what face will be uppermost until we roll the die.

However,
the *sample space* for this procedure can be listed:
**1**, **2**, **3**, **4**, **5**, **6**.
These are all distinct results (no overlap),
and the sample space is
*discrete*.

Many *events* could be defined using this sample space; for example:

- Rolling a
**4**: This event includes one element of the sample space:**4**. - Rolling a even number: This event includes three elements of the sample space:
**2**,**4**and**6**. - Rolling a number larger than
**2**: This event includes four elements of the sample space:**3**,**4**,**5**and**6**.

**Example 16.2 (Sample spaces and events) **Consider the distance which you can throw a cricket ball.
We do not know what distance your throw will be until you throw,
but we can describe the *sample space* for this procedure:
the distance could be anywhere between (say) 0 and 200 metres
(and of course, some of those distances are fairly unlikely to occur…).

This sample space is
*continuous*.

Many *events* could be defined using this sample space; for example:

- Throwing more than 50 metres: This event includes elements of the sample space greater than 50m.
- Throwing between 10 and 40 metres: This event includes elements of the sample space between 10m and 40m.
- Throwing less than 20 metres: This event includes elements of the sample space less than 20m.

*exact*distance (such as

*exactly*10 metres) is technically not possible.

Once a sample space is defined,
a *probability* can be defined.

**Definition 16.3 (Probability)**A

*probability*is a number between 0 and 1 inclusive (or between 0% and 100% inclusive) that quantifies the likelihood that a certain

*event*will occur.

A probability of zero (or 0%) means the event is ‘impossible’ (will never occur). At the other extreme, a probability of one (or 100%) means that the event is ‘certain’ to happen (will always occur). Most events have a probability between the extremes of 0% and 100%.

**Example 16.3 (Probabilities) **Consider these cases:

- The probability of receiving negative rainfall is zero: It is impossible.
- The probability of receiving some rain in Buderim in the next decade is one. It is certain.
- The probability of receiving rain on any given day… is somewhere between 0 and 1.

Different ways exist to calculate probabilities of events, including: