NSW Syllabuses

# Mathematics Advanced Stage 6 - Year 11 - Statistical Analysis MA-S2 Probability

## Outcomes

#### A student:

• MA11-7

uses concepts and techniques from statistics and probability to present and interpret data and solve problems in a variety of contexts, including the use of probability distributions

• MA11-8

uses appropriate technology to investigate, organise, model and interpret information in a range of contexts

• MA11-9

provides reasoning to support conclusions which are appropriate to the context

## Subtopic Focus

The principal focus of this subtopic is a review of fundamentals of probability and introduction of concepts of conditional probability and independence.

Students develop skills related to probability, its language and visual representations, and use these to solve practical problems.

## Content

• Students:
• review, understand and use the language associated with theoretical probability and relative frequency (ACMMM049)
• construct a sample space for an experiment and use it to determine the number of possible outcomes (ACMEM154, ACMEM155)
• review probability as a measure of the likely chance of occurrence of an event
• review the probability scale: $$0\le P(A)\le1$$ for an event $$A$$, with $$P(A)=0$$ if $$A$$ is an impossibility and $$P(A)=1$$ if $$A$$ is a certainty (ACMMM053)
• solve problems involving relative frequency in a variety of contexts
• perform simulations of experiments using digital technology (ACMEM150)
• use relative frequency as an estimate of probability (ACMEM152)
• recognise that an increasing number of trials produces relative frequencies that become closer in value to the theoretical probability (ACMEM151)
• recognise that in some situations it may not be possible to calculate a theoretical probability, for example the probability of a drawing pin landing on its side when thrown or the probability that a person chosen at random has a particular disease
• identify factors that could complicate the simulation of real-world events (ACMEM153)
• use Venn diagrams, set language and notation for events, including $$\bar A$$ (or $$A^c$$) for the complement of an event $$A$$, $$A\cap B$$ for the intersection of events $$A$$ and $$B$$, and $$A\cup B$$ for the union of events $$A$$ and $$B$$ (ACMMM050)
• recognise that for mutually exclusive events $$P(A \cap B) = 0$$ (since $$A\cap B = \emptyset$$ where $$\emptyset$$ is the empty set and hence $$P(A\cup B)=P(A)+P(B)$$ (ACMMM054)
• use everyday occurrences to illustrate set descriptions and representations of events and set operations (ACMMM051)
• use Venn diagrams to illustrate and interpret simple probability situations where appropriate
• review and use the rules: $$P\left(\bar{A}\right) = 1- P(A)$$ and $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$   (ACMMM054)
• use the notation $$P(A|B)$$ and the formula $$P(A|B)=\frac{P(A\cap B)}{P(B)}$$ for conditional probability (ACMMM057)
• understand the notion of conditional probability in terms of reduced sample space and recognise and use language that indicates conditionality (ACMMM056)
• use the multiplication law $$P(A \cap B) = P(A)P(B)$$ for independent events $$A$$ and $$B$$ and recognise the symmetry of independence in simple probability situations (ACMMM059)
• use arrays and tree diagrams to determine the outcomes and probabilities for multi-stage experiments (ACMEM156)
• construct and use tree diagrams to establish the outcomes for a simple multi-stage event
• use probability tree diagrams to solve problems involving two-stage events