The position of an object on a plane or in space can be specified and described relative to a reference point.

Experimental probability is calculated by the frequency of an event occurring based on repeated trials.

Randomness is the unpredictability of an outcome occurring. It is not possible to predict which outcome in a trial will occur because randomness is not influenced by any factor other than chance.

Outcomes are fair when there is an equal chance of occurrence.

Biased outcomes do not have an equal chance of occurrence. They are not fair.

An independent event is an event that is not affected by the outcome of another event.

A dependent event is an event that is affected by the outcome of a prior event.

The probability of events can be described using language and/or numerical terms.

Language descriptions

Words can be used to describe and order probabilities.

Numerical descriptions

Numbers from zero to one can be used to express probabilities. A probability of zero describes an event that is impossible to occur. A probability of one describes an event that is certain to occur. A probability of one-half describes an event that has the same chance of occurring as the chance of it not occurring.

Ratios are sometimes used to represent the odds of an event occurring.

Experiment

An experiment is an enactment of a situation.

Trial

A trial is a particular performance of an experiment.

Outcome

An outcome is one possible result for a single trial.

Frequency

Frequency refers to the number of times an outcome occurs.

Sample space

The sample space is the set of all the possible

outcomes of an experiment.

Event

An event is a subset of the sample space for a

random experiment (ACARA 2019).

There are several ways to systematically determine the number of possible outcomes (i.e. sample space) for situations involving elements of chance.

Consider the situation of a game played with a paper cup containing four marbles (two white and two black). Two marbles are drawn out. The following are representations of the possible outcomes.

Systematic list

In a systematic list, the most important step is to label each marble individually then match the marbles into pairs systematically. That way, no possible pairings are missed. Note that the order of marbles coming out is considered:

• W1W2, W1B1, W1B2

• W2W1, W2B1, W2B2

• B1W1, B1W2, B1B2

• B2W1, B2W2, B2B1.

Tree diagram

In a tree diagram, the end of each arm is a different outcome (see Figure 62). Note that each marble has an individual code.

In a table (sometimes called a matrix), each cell represents a different outcome.

Subjective judgements

Students often describe the chance of an event in terms of ‘personal feelings’.

Recency effect

Students can mistakenly make predictions about the likelihood of an independent event based on the outcome of the previous trials.

Independence effect

Students might mistakenly make predictions about the likelihood of an independent event based on the outcome of the previous trials. Chance has no memory.

Sampling variability effect

Variations in the results of an experiment due to sampling often cause students to question their theoretical model.

Equi-probability bias

Students can mistakenly believe that every chance situation is fair and has equal probabilities. However, in more complex situations, events often have unequal probabilities.