Making Predictions… · SHS2 Additional Mathematics

Lesson notes

Learning Objectives

Indicator: 2.4.2.LI.2 - Investigate the axioms of probability: For any event A, there is 0 ≤ P(A) ≤ 1; that is, the probability of an event A is a number between 0 and 1 inclusive.

By the end of the lesson, learners can:

State all three axioms of probability correctly: P(S) = 1, P(∅) = 0, and 0 ≤ P(A) ≤ 1 for any event A.
Use the complement rule P(A’) = 1 - P(A) to find the probability of an event not happening.
Identify which axiom applies to a given event and justify their choice orally in group discussion.
Solve problems involving probabilities of independent events using the multiplication law and the complement rule.
Interpret probability values in real-life situations and explain what a probability of 0, 1, or a value in between means in plain language.

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Curriculum details

Strand: Handling Data (Strand 4)
Sub-strand: Making Predictions with Data (4.2)
Content standard: 2.4.2.CS.1 - Apply the addition and multiplication laws and axioms of probability to solve real life problems. 2.4.2.LO.1 Solve problems using the axioms and the laws of probability. 2.4.2.LO.2 Solve real life problems using combination and permutation.
Indicator: 2.4.2.LI.2 - Investigate the axioms of probability: For any event A, there is 0 ≤ 𝑃(𝐴) ≤ 1; that is, the probability of an event A is a number between 0 and 1 inclusive.
Suggested placement: Semester 2, Week 17 (Week 37 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

indicator text - p.422: the page prints 2.4.2.AS.2 in the learning-indicator column as well as the assessment column. Filed as 2.4.2.LI.2, the number its position under 2.4.2.CS.1 gives it, since an assessment code cannot identify an indicator
Curriculum reference: NaCCA curriculum document, p. 422

Exemplars (from the NaCCA curriculum)

Group work, Talk for Learning, and Building on what others say.
Learning Experience: Learners in groups investigate axioms of probability.
Activity 1 Properties and axioms of probability Learners brainstorm in groups to investigate in an experiment the properties and axioms of probability. Thus, for any outcome or event, 𝐴 of an experiment;
- 𝑃(𝑆) = 1, where 𝐴 = 𝑆, the sample space.
- 𝑃(∅) = 0, 𝐴 = ∅, 𝐴 is a null set
- 0 ≤ 𝑃(𝐴) ≤ 1, where 𝐴 is a proper of subsets of 𝑆
- a) 𝑃(𝐴) = 1 − 𝑃(𝐴 - ) b) 𝑃(𝐴 - ) = 1 = 𝑃(𝐴)
Example: The probabilities of Ben and Alice solving a mathematics problem correctly are 0.7 and 0.8, respectively. Find the probability that
- only Alice solves it correctly.
- both Ben and Alice fail to solve it.
- only one of them solves it correctly.
- What is the probability of at least one of them solving the problem correctly?
Solution Let 𝑃(Alice solving correctly)= 𝑃(𝐴) and 𝑃(Ben solving correctly) = 𝑃(𝐵) 𝑃(𝐴) = 0.8, 𝑃(𝐴 - ) = 0.2 𝑃(𝐵) = 0.7, 𝑃(𝐵 - ) = 0.3 a) 𝑃(only Alice Solving) =𝑃(𝐴) × 𝑃(𝐵 - ) = 0.8 × 0.3 = 0.24 b) 𝑃(both failed to solve)= 𝑃(𝐵 - ) × 𝑃(𝐴 - ) = 0.3 × 0.2 = 0.06 c) 𝑃(only one solved it)= 𝑃(𝐴) ∙ 𝑃(𝐵 - ) + 𝑃(𝐴 - ) ∙ 𝑃(𝐵) = 0.8 × 0.3 + 0.2 × 0.7 = 0.24 + 0.14
= 0.38 d) 𝑃(at least one of them solving it correctly)= 1 − 𝑃(none solving) = 1 − 𝑃(None solving) = 1 − 𝑃(𝐴 - ) ∙ 𝑃(𝐵 - ) = 1 − 0.06 = 0.94
Teaching and Learning Resources:
- Real-life data samples from both the school community and/or outside the school community.
- ICT tools
Assessment (2.4.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding.