NSW Syllabuses

# Mathematics K–10 - Stage 3 - Statistics and Probability Chance

## Chance 1

### Outcomes

#### A student:

• MA3-1WM

describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions

• MA3-19SP

conducts chance experiments and assigns probabilities as values between 0 and 1 to describe their outcomes

### Content

• Students:
• List outcomes of chance experiments involving equally likely outcomes and represent probabilities of those outcomes using fractions (ACMSP116)
• use the term 'probability' to describe the numerical value that represents the likelihood of an outcome of a chance experiment
• recognise that outcomes are described as 'equally likely' when any one outcome has the same chance of occurring as any other outcome
• list all outcomes in chance experiments where each outcome is equally likely to occur
• represent probabilities of outcomes of chance experiments using fractions, eg for one throw of a standard six-sided die or for one spin of an eight-sector spinner
• determine the likelihood of winning simple games by considering the number of possible outcomes, eg in a 'rock-paper-scissors' game (Problem Solving, Reasoning)
• Recognise that probabilities range from 0 to 1 (ACMSP117)
• establish that the sum of the probabilities of the outcomes of any chance experiment is equal to 1
• order commonly used chance words on an interval from zero ('impossible') to one ('certain'), eg 'equally likely' would be placed at $$\frac{1}{2}$$ (or 0.5)
• describe events that are impossible and events that are certain (Communicating)
• describe the likelihood of a variety of events as being more or less than a half (or 0.5) and order the events on an interval (Communicating)

### Background Information

Students will need some prior experience in ordering fractions and decimals on a number line from 0 to 1.

The probability of chance events occurring can be ordered on a scale from 0 to 1. A probability of 0 describes the probability of an event that is impossible. A probability of 1 describes the probability of an event that is certain. Events with an equal likelihood of occurring or not occurring can be described as having a probability of $$\frac{1}{2}$$ (or 0.5 or 50%). Other expressions of probability fall between 0 and 1, eg events described as 'unlikely' will have a numerical value somewhere between 0 and $$\frac{1}{2}$$ (or 0.5 or 50%).

The sum of the probabilities of the outcomes of any chance experiment is equal to 1. This can be demonstrated by adding the probabilities of all of the outcomes of a chance experiment, such as rolling a die.

### Language

Students should be able to communicate using the following language: chance, event, likelihood, certain, possible, likely, unlikely, impossible, experiment, outcome, probability.

The probability of an outcome is the value (between 0 and 1) used to describe the chance that the outcome will occur.

A list of all of the outcomes for a chance experiment is known as the 'sample space'; however, this term is not introduced until Stage 4.

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA3‑19SP the sub-elements (and levels) of Comparing units (CoU1-CoU2) and Understanding chance (UnC3-UnC5) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.

## Chance 2

### Outcomes

#### A student:

• MA3-1WM

describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions

• MA3-2WM

selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations

• MA3-3WM

gives a valid reason for supporting one possible solution over another

• MA3-19SP

conducts chance experiments and assigns probabilities as values between 0 and 1 to describe their outcomes

### Content

• Students:
• Compare observed frequencies across experiments with expected frequencies (ACMSP146)
• use the term 'frequency' to describe the number of times a particular outcome occurs in a chance experiment
• distinguish between the 'frequency' of an outcome and the 'probability' of an outcome in a chance experiment (Communicating)
• compare the expected frequencies of outcomes of chance experiments with observed frequencies, including where the outcomes are not equally likely
• recognise that some random generators have outcomes that are not equally likely and discuss the effect on expected outcomes, eg on this spinner, green is more likely to occur than red or grey or blue
(Reasoning)
• discuss the 'fairness' of simple games involving chance (Communicating, Reasoning)
• explain why observed frequencies of outcomes in chance experiments may differ from expected frequencies (Communicating, Reasoning)
• list the outcomes for chance experiments where the outcomes are not equally likely to occur and assign probabilities to the outcomes using fractions
• use knowledge of equivalent fractions, decimals and percentages to assign probabilities to the likelihood of outcomes, eg there is a 'five in ten', $$\frac{1}{2}$$, 50%, 0.5 or 'one in two' chance of a particular event occurring
• use probabilities in real-life contexts, eg 'My football team has a 50% chance of winning the game' (Communicating, Reasoning)
• design a spinner or label a die so that a particular outcome is more likely than another and discuss the probabilities of the outcomes (Communicating, Problem Solving)
• Conduct chance experiments with both small and large numbers of trials using appropriate digital technologies (ACMSP145)
• assign expected probabilities to outcomes in chance experiments with random generators, including digital simulators, and compare the expected probabilities with the observed probabilities after both small and large numbers of trials
• determine and discuss the differences between the expected probabilities and the observed probabilities after both small and large numbers of trials (Communicating, Reasoning)
• explain what happens to the observed probabilities as the number of trials increases (Communicating, Reasoning)
• use samples to make predictions about a larger 'population' from which the sample comes, eg take a random sample of coloured lollies from a bag, calculate the probability of obtaining each colour of lolly when drawing a lolly from the bag, and use these probabilities and the total number of lollies in the bag to predict the number of each colour of lolly in the bag
• discuss whether a prediction about a larger population, from which a sample comes, would be the same if a different sample were used (Communicating, Reasoning)

### Background Information

Random generators include coins, dice, spinners and digital simulators.

As the number of trials in a chance experiment increases, the observed probabilities should become closer in value to the expected probabilities.

Refer also to background information in Chance 1.

### Language

Students should be able to communicate using the following language: chance, event, likelihood, equally likely, experiment, outcome, expected outcomes, random, fair, trials, probability, expected probability, observed probability, frequency, expected frequency, observed frequency.

The term 'frequency' is used in this substrand to describe the number of times a particular outcome occurs in a chance experiment. In Stage 4, students will also use 'frequency' to describe the number of times a particular data value occurs in a data set.

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA3‑19SP the sub-elements (and levels) of Operating with decimals (OwD3), Operating with percentages (OwP1-OwP2), Comparing units (CoU1-CoU2), Understanding chance (UnC2-UnC5) and Interpreting and representing data (IRD5) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.