Outcomes
A student:

 MA5.21WM
selects appropriate notations and conventions to communicate mathematical ideas and solutions

 MA5.22WM
interprets mathematical or reallife situations, systematically applying appropriate strategies to solve problems

 MA5.23WM
constructs arguments to prove and justify results

 MA5.217SP
describes and calculates probabilities in multistep chance experiments
Related Life Skills outcomes: MALS38SP, MALS39SP
Content
 Students:
 List all outcomes for twostep 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 twostep 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 twostep chance experiments, with and without replacement, using organised lists, tables and tree diagrams
 calculate probabilities of simple and compound events in twostep chance experiments, with and without replacement
 explain the effect of knowing the result of the first step on the probability of events in twostep chance experiments, with and without replacement (Communicating, Reasoning)
 Describe the results of two and threestep chance experiments, with and without replacement, assign probabilities to outcomes, and determine probabilities of events; investigate the concept of independence (ACMSP246)
 distinguish informally between dependent and independent events
 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 threestep chance experiments, eg draw three counters from a bag containing three blue, four red and one white counter
 record outcomes of threestep chance experiments, with and without replacement, using organised lists, tables and tree diagrams
 calculate probabilities of simple and compound events in threestep chance experiments, with and without replacement
 use knowledge of complementary events to assist in calculating probabilities of events in multistep 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 sixsided 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 sixsided die. He felt that he should also win betting on obtaining at least one double 6 in 24 rolls of two fair sixsided 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 sixsided 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 MA5.217SP the subelements (and levels) of Understanding chance (UnC5) describe observable behaviours that can aid teachers in making evidencebased decisions about student development and future learning.
The progression subelements and indicators can be viewed by accessing the National Numeracy Learning Progression.