Australian Curriculum Mathematics/Mathematical Methods/Counting and Probability

Australian Curriculum Content

Combinations

• understand the notion of a combination as an unordered set of r objects taken from a set of n distinct objects

• use the notation (nr) and the formula (nr)=n!/(n−r)! for the number of combinations of r objects taken from a set of n distinct objects

• expand (x+y)^n for small positive integers n

• recognise the numbers (nr) as binomial coefficients, (as coefficients in the expansion of (x+y)^n)

• use Pascal’s triangle and its properties.

Language of events and sets

• review the concepts and language of outcomes, sample spaces and events as sets of outcomes

• use set language and notation for events, including A¯¯¯ (or A') for the complement of an event A, A∩B for the intersection of events A and B, and A?B for the union, and recognise mutually exclusive events

• use everyday occurrences to illustrate set descriptions and representations of events, and set operations.

Review of the fundamentals of probability

• review probability as a measure of ‘the likelihood of occurrence’ of an event

• review the probability scale: 0≤P(A)≤1 for each event A, with P(A)=0 if A is an impossibility and P(A)=1 if A is a certainty

• review the rules: P(A¯¯¯)=1−P(A) and P(A∪B)=P(A)+P(B)−P(A∩B)

• use relative frequencies obtained from data as point estimates of probabilities.

Conditional probability and independence

• understand the notion of a conditional probability and recognise and use language that indicates conditionality

• use the notation P(A|B) and the formula P(A∩B)=P(A|B)P(B)

• understand the notion of independence of an event A from an event B, as defined by P(A|B)=P(A)

• establish and use the formula P(A∩B)=P(A)P(B) for independent events A and B, and recognise the symmetry of independence

• use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events.