ADVANCED PROBABILITY QUESTION about multi-stage event probability and step by step solution in two different ways (methods). Students think critically to solve the “Probability of Winning Both Games” question. This challenge question requires students to use their knowledge & understanding of probability concepts and to come up with mathematical strategies to solve the problem. Learning Outcomes: • Use arrays or tree diagrams to determine the outcomes and the probabilities for multi-stage events. • Understand events using the language of ‘equally likely’, ‘at least’, etc. • Use complementary event to determine the probabilities associated with simple games. • Use & solve a quadratic equation form that related to the probability of multi-stage events. Also comply with Australian Curriculum Identify complementary events and use the sum of probabilities to solve problems (ACMSP204) 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) Describe the results of two- and three-step chance experiments, both with and without replacement, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence (ACMSP246) Simple probabilities: construct a sample space for an experiment (ACMEM154) Use a sample space to determine the probability of outcomes for an experiment (ACMEM155) Use arrays or tree diagrams to determine the outcomes and the probabilities for experiments. (ACMEM156) Probability applications: determine the probabilities associated with simple games (ACMEM157) Probability

PROBABILITY REVISION 2 WORKSHEETS AND STEP-BY-STEP SOLUTIONS Learning Outcomes: • Venn diagrams and the addition theorem • Multi-stage experiments and the product rule • Mutually Exclusive Events • Probability tree diagrams • Complementary Events Also comply with Australian Curriculum Review the concepts and language of outcomes, sample spaces and events as sets of outcomes (ACMMM049) Use set language and notation for events, including A` for the complement of an event A, A∩B for the intersection of events A and B, and AՍB for the union, and recognise mutually exclusive events (ACMMM050) Use everyday occurrences to illustrate set descriptions and representations of events, and set operations. (ACMMM051) Review probability as a measure of ‘the likelihood of occurrence’ of an event (ACMMM052) 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 certain (ACMMM053) Review the rules: P(A`) = 1 − P(A) and P(A∪B) = P(A) + P(B) − P(A∩B) (ACMMM054)

**PROBABILITY REVISION 1 WORKSHEETS AND STEP-BY-STEP SOLUTIONS** Learning Outcomes: • Venn diagrams and the addition theorem • Multi-stage experiments and the product rule • Mutually Exclusive Events • Probability tree diagrams • Conditional Probability • Independent Events Also comply with Australian Curriculum Review the concepts and language of outcomes, sample spaces and events as sets of outcomes (ACMMM049) Use set language and notation for events, including A` for the complement of an event A, A∩B for the intersection of events A and B, and AՍB for the union, and recognise mutually exclusive events (ACMMM050) Use everyday occurrences to illustrate set descriptions and representations of events, and set operations. (ACMMM051) Review probability as a measure of ‘the likelihood of occurrence’ of an event (ACMMM052) 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 certain (ACMMM053) Review the rules: P(A`) = 1 − P(A) and P(A∪B) = P(A) + P(B) − P(A∩B) (ACMMM054) Use relative frequencies obtained from data as point estimates of probabilities. (ACMMM055) Understand the notion of a conditional probability and recognise and use language that indicates conditionality (ACMMM056) Use the notation P(A|B) and the formula P(A|B) = P(A∩B)/P(B) (ACMMM057) Understand the notion of independence of an event A from an event B, as defined by P(A|B) = P(A) (ACMMM058) Establish and use the formula P(A∩B) = P(A)P(B) for independent events A and B, and recognise the symmetry of independence (ACMMM059) Use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events. (ACMMM060)