NSW Syllabuses

# Mathematics K–10 - Stage 5.2 - Statistics and Probability Probability ◊

## Outcomes

#### A student:

• MA5.2-1WM

selects appropriate notations and conventions to communicate mathematical ideas and solutions

• MA5.2-2WM

interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems

• MA5.2-3WM

constructs arguments to prove and justify results

• MA5.2-17SP

describes and calculates probabilities in multi-step chance experiments

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

## Content

• Students:
• List all outcomes for two-step chance experiments, with and without replacement, using tree diagrams or arrays; assign probabilities to outcomes and determine probabilities for events (ACMSP225)
• sample, with and without replacement, in two-step chance experiments, eg draw two counters from a bag containing three blue, four red and one white counter
• compare results between an experiment undertaken with replacement and then without replacement (Reasoning)
• record outcomes of two-step chance experiments, with and without replacement, using organised lists, tables and tree diagrams
• calculate probabilities of simple and compound events in two-step chance experiments, with and without replacement
• explain the effect of knowing the result of the first step on the probability of events in two-step chance experiments, with and without replacement (Communicating, Reasoning)
• Describe the results of two- and three-step chance experiments, with and without replacement, assign probabilities to outcomes, and determine probabilities of events; investigate the concept of independence (ACMSP246)
• explain the difference between dependent and independent events using appropriate examples (Communicating, Reasoning)
• recognise that for independent events $$P \left( A \,\mbox { and } \, B \right) = P \left( A \right) \times P \left( B \right)$$
• sample, with and without replacement, in three-step chance experiments, eg draw three counters from a bag containing three blue, four red and one white counter
• record outcomes of three-step chance experiments, with and without replacement, using organised lists, tables and tree diagrams
• calculate probabilities of simple and compound events in three-step chance experiments, with and without replacement
• use knowledge of complementary events to assist in calculating probabilities of events in multi-step chance experiments (Problem Solving)
• evaluate the likelihood of winning a prize in lotteries and other competitions (Problem Solving, Reasoning)
• Use the language of 'if ... then', 'given', 'of', 'knowing that' to investigate conditional statements and to identify common mistakes in interpreting such language (ACMSP247)
• calculate probabilities of events where a condition is given that restricts the sample space, eg given that a number less than 5 has been rolled on a fair six-sided die, calculate the probability that this number was a 3
• describe the effect of a given condition on the sample space, eg in the above example, the sample space is reduced to {1,2,3,4} (Communicating, Problem Solving, Reasoning)
• critically evaluate conditional statements used in descriptions of chance situations
• describe the validity of conditional statements used in descriptions of chance situations with reference to dependent and independent events, eg explain why if you toss a coin and obtain a head, then the probability of obtaining a head on the next toss remains the same (Communicating, Reasoning)
• identify and explain common misconceptions related to chance experiments, eg explain why the statement 'If you obtain a tail on each of four consecutive tosses of a coin, then there is a greater chance of obtaining a head on the next toss' is incorrect (Reasoning)

### Background Information

Meteorologists use probability to predict the weather and to communicate their predictions, eg 'There is a 50% chance of rain tomorrow'. Insurance companies use probability to determine premiums, eg the chance of particular age groups having accidents.

The mathematical analysis of probability was prompted by the French writer and gambler Antoine Gombaud, the Chevalier de Méré (1607–1684). Over the years, the Chevalier had consistently won money betting on obtaining at least one 6 in four rolls of a fair six-sided die. He felt that he should also win betting on obtaining at least one double 6 in 24 rolls of two fair six-sided dice, but in fact regularly lost.

In 1654 he asked his friend Blaise Pascal (1623–1662), the French mathematician and philosopher, to explain why he regularly lost in the second situation. This question led to the famous correspondence between Pascal and the renowned French lawyer and mathematician Pierre de Fermat (1601–1665). Chevalier's losses are explained by the fact that the chance of obtaining at least one 6 in four rolls of a die is $$\, 1- \left( \frac{5}{6} \right)^{\!4} \!\approx 51.8 \% \,\, , \,$$ while the chance of obtaining at least one double 6 in 24 rolls of two dice is $$\, 1 - \left( \frac{35}{36} \right) ^{24} \!\approx 49.1 \% \,\,$$.

### Language

In a chance experiment, such as rolling a fair six-sided die twice, an event is a collection of outcomes. For instance, an event in this situation might be that the result is 'a sum of 7' or 'a sum of 10 or more'.

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

When working towards the outcome MA5.2-17SP 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.