NSW Syllabuses

# Mathematics K–10 - Stage 4 - Statistics and Probability Probability

## Probability 1

### Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-3WM

recognises and explains mathematical relationships using reasoning

• MA4-21SP

represents probabilities of simple and compound events

Related Life Skills outcomes: MALS-38SP, MALS-39SP

• use the term 'chance experiment' when referring to actions such as tossing a coin, rolling a die, or randomly selecting an object from a bag
• use the term 'outcome' to describe a possible result of a chance experiment and list all of the possible outcomes for a single-step experiment
• use the term 'sample space' to describe a list of all of the possible outcomes for a chance experiment, eg if a standard six-sided die is rolled once, the sample space is {1,2,3,4,5,6}
• distinguish between equally likely outcomes and outcomes that are not equally likely in single-step chance experiments
• describe single-step chance experiments in which the outcomes are equally likely, eg the outcomes for a single toss of a fair coin (Communicating, Reasoning)
• describe single-step chance experiments in which the outcomes are not equally likely, eg the outcomes for a single roll of a die with six faces labelled 1, 2, 3, 4, 4, 4 are not equally likely since the outcome '4' is three times more likely to occur than any other outcome (Communicating, Reasoning)
• design a spinner, given the relationships between the likelihood of each outcome, eg design a spinner with three colours, red, white and blue, so that red is twice as likely to occur as blue, and blue is three times more likely to occur than white (Problem Solving)
• Assign probabilities to the outcomes of events and determine probabilities for events (ACMSP168)
• use the term 'event' to describe either one outcome or a collection of outcomes in the sample space of a chance experiment, eg in the experiment of rolling a standard six-sided die once, obtaining the number '1' is an 'event' and obtaining a number divisible by three is also an event
• explain the difference between experiments, events, outcomes and the sample space in chance situations (Communicating)
• assign a probability of 0 to events that are impossible and a probability of 1 to events that are certain to occur
• explain the meaning of the probabilities 0, $$\frac{1}{2}$$ and 1 in a given chance situation (Communicating)
• assign probabilities to simple events by reasoning about equally likely outcomes, eg the probability of randomly drawing a card of the diamond suit from a standard pack of 52 playing cards is $$\frac{13}{52} = \frac{1}{4}$$
• express the probability of an event, given a finite number of equally likely outcomes in the sample space, as $$P\left(\mbox{event}\right) = \frac{\mbox{number of favourable outcomes}}{\mbox{total number of outcomes}}$$
• solve probability problems involving single-step experiments using cards, dice, spinners, etc
• establish that the sum of the probabilities of all of the possible outcomes of a single-step experiment is 1
• identify and describe the complement of an event, eg the complement of the event 'rolling a 6' when rolling a die is 'not rolling a 6'
• establish that the sum of the probability of an event and its complement is 1, ie $$P\left(\text{event}\right) + P\left(\text{complement of event}\right) = 1$$
• calculate the probability of a complementary event using the fact that the sum of the probabilities of complementary events is 1, eg the probability of 'rolling a 6' when rolling a die is $$\frac{1}{6}$$, therefore the probability of the complementary event, 'not rolling a 6', is $$1- \frac{1}{6} = \frac{5}{6}$$

#### Purpose/Relevance of Substrand

Probability is concerned with the level of certainty that a particular event will occur. The higher the probability of an event, the 'more certain' or 'more likely' it is that the event will occur. Probability is used widely by governments and in many fields, including mathematics, statistics, science, business and economics. In everyday situations, probabilities are key to such areas as risk assessment, finance, and the reliability of products such as cars and electronic goods. It is therefore important across society that probabilities are understood and used appropriately in decision making.

### Language

A simple event has outcomes that are equally likely. In a chance experiment, such as rolling a standard six-sided die once, an event might be one of the outcomes or a collection of the outcomes. For example, an event might be that an odd number is rolled, with the favourable outcomes being a '1', a '3' and a '5'.

It is important that students learn the correct terminology associated with probability.

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

When working towards the outcome MA4‑21MG the sub-elements (and levels) of Operating with percentages (OwP3), Comparing units (CoU2), Interpreting fractions (InF5) 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.

## Probability 2

### Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-3WM

recognises and explains mathematical relationships using reasoning

• MA4-21SP

represents probabilities of simple and compound events

Related Life Skills outcomes: MALS-38SP, MALS-39SP

• Students:
• Describe events using language of 'at least', exclusive 'or' (A or B but not both), inclusive 'or' (A or B or both) and 'and' (ACMSP205)
• recognise the difference between mutually exclusive and non-mutually exclusive events, eg when a die is rolled, 'rolling an odd number' and 'rolling an even number' are mutually exclusive events; however, 'rolling an even number' and 'rolling a 2' are non-mutually exclusive events
• describe compound events using the following terms:
• 'at least', eg rolling a 4, 5 or 6 on a standard six-sided die may be described as rolling 'at least 4'
• 'at most', eg rolling a 1, 2, 3 or 4 on a standard six-sided die may be described as rolling 'at most 4'
• 'not', eg choosing a black card from a standard pack of cards may be described as choosing a card that is 'not red'
• 'and', eg choosing a card that is black and a king means that the card must have both attributes
• pose problems that involve the use of these terms, and solve problems posed by others (Communicating, Problem Solving)
• describe the effect of the use of 'and' and 'or' when using internet search engines (Communicating, Problem Solving)
• classify compound events using inclusive 'or' and exclusive 'or', eg 'choosing a male or a female' is exclusive as one cannot be both, whereas 'choosing a male or someone left-handed' could imply exclusivity or inclusivity
• recognise that the word 'or' on its own often needs a qualifier, such as 'both' or 'not both', to determine inclusivity or exclusivity (Reasoning)
• interpret Venn diagrams involving two or three attributes
• describe regions in Venn diagrams representing mutually exclusive attributes, eg a Venn diagram representing the languages studied by Year 8 students

There are 50 students who study French; 32 students who study Mandarin; 18 students who study neither language; and no student who studies both languages (Communicating, Problem Solving, Reasoning)
• describe individual regions or combinations of regions in Venn diagrams representing non-mutually exclusive attributes, using the language 'and', exclusive 'or', inclusive 'or', 'neither' and 'not', eg a Venn diagram representing the sports played by Year 8 students

There are 25 students who play both basketball and football; 46 students who play basketball or football, but not both; 19 students who play neither sport; and 71 students who play basketball or football or both (Communicating, Problem Solving, Reasoning)
• construct Venn diagrams to represent all possible combinations of two attributes from given or collected data
• use given data to calculate missing values in a Venn diagram, eg the number of members that have both attributes or the number of members that have neither attribute (Problem Solving, Reasoning)
• interpret given two-way tables representing non-mutually exclusive attributes
• describe relationships displayed in two-way tables using the language 'and', exclusive 'or', inclusive 'or', 'neither' and 'not', eg a table comparing gender and handedness of students in Year 8

There are 63 male right-handed students, ie 63 students are neither female nor left-handed; there are 114 students who are male or right-handed, or both (Communicating, Problem Solving, Reasoning)
• construct two-way tables to represent the relationships between attributes
• use given data to calculate missing values in a two-way table (Problem Solving)
• convert between representations of the relationships between two attributes in Venn diagrams and two-way tables, eg

### Background Information

John Venn (1834−1923) was a British mathematician best known for his diagrammatic way of representing sets, and their unions and intersections.

Students are expected to be able to interpret Venn diagrams involving three attributes; however, they are not expected to construct Venn diagrams involving three attributes.

### Language

A compound event is an event that can be expressed as a combination of simple events, eg drawing a card that is black or a King from a standard set of playing cards, throwing at least 5 on a standard six-sided die.

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

When working towards the outcome MA4‑21MG the sub-elements (and levels) of Understanding chance (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.