SHS1 Mathematics · Semester 2 Week 19: Probability/chance

Lesson notes

Learning Objectives

Indicator: 1.4.2.LI.2 Determine the probabilities of independent events and express the results as fractions, decimals, percentages and/or ratios.

By the end of the lesson, learners can:

State the multiplication rule for independent events and explain why P(A and B) = P(A) × P(B) when A and B are independent.
Compute the probability of two independent events occurring together using the multiplication rule, given the individual probabilities.
Use the complement rule (P(at least one) = 1 - P(none)) to find probabilities of combined independent events.
Express probability answers in at least three different forms: as a fraction, decimal, percentage, and/or ratio.
Apply the multiplication rule to solve simple problems involving two independent events in everyday situations.

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Curriculum details

Strand: Making Sense of and Using Data (Strand 4)
Sub-strand: Probability/chance (4.2)
Content standard: 1.4.2.CS.1 - Demonstrate conceptual understanding of simple and compound probability experiments involving two independent events. 1.4.2.LO.1 Determine the sample space for simple and compound probability experiments involving independent events; express the probabilities of given events as fractions, decimals, percentages and/or ratios and solve problems everyday life problems.
Indicator: 1.4.2.LI.2 - Determine the probabilities of independent events and express the results as fractions, decimals, percentages and/or ratios.
Suggested placement: Semester 2, Week 19 (Week 39 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:

exemplars - p.151: this exemplar sets a two-dimensional construct - a stacked fraction, an index or a column vector - which the text layer flattens into separate lines, so the parts are present but their vertical arrangement is lost; read the page
Curriculum reference: NaCCA curriculum document, p. 151

Exemplars (from the NaCCA curriculum)

Using think-pair-share activities, engage learners to discuss and solve problems on probabilities of independent events.
Example 1: Let us suppose there are ten balls in a bo. Four balls are Green (G), and six balls are Red(R). If we draw two balls, one at a time, with replacement, find the probability of the following events: 1. Both Balls are Green. 2. The first ball is Red, and the second is Green. 3. At least one ball is Red.
Solution: Let G1 and R1 be the events that the first ball is Green/Red, respectively. Similarly, let G2 and R2 be the events that the second ball is Green/Red. Since we are dealing with sampling with replacement, so 4 2 6 3 P(G1) = P(G2) = = and P(R1) = P(R2) = = 10 5 10 5
1. P(both balls are green) = P(G1 and G2) = P(G1∩ G2). Since the trials are independent, so P(G1∩ 2 2 4 G2) = P(G1) × P(G2) = 5 × 5 = 2 .
2. P(first Red and Second Green) = P(R1 and G2) = P(R1 ∩ 𝐺2). since the trials are independent, so 3 2 6 P(R1 ∩ 𝐺2)= P(R1) × P(G2) = 5 × 5 = 25 .
3. We use the fact that P(at least one ball is Red) = 1 - P(both balls are Green). Hence, P(at least 4 21 one ball is Red) = 1 − = . 25 25
Example 2: A poll finds that 72% of the youth in Gushegu consider themselves football fans. If you randomly pick two people from the population, what is the probability that
- the first person is a football fan and the second is as well.
- the first one is, and the second one isn't?
Solution: One person being a football fan does not have an effect on whether the second randomly selected person is. Therefore, the events are independent, and the probability can be found by multiplying the probabilities together: - First and second are football fans: P(A∩B) = P(A) · P(B) = .72 * .72 = .5184. - First one is a football fan, the second one isn't: P(A∩B) = P(A) · P(B) = .72 * 1 - 0.72) = 0.202. In the second part, I multiplied by the complement. As the probability of being a fan is .72, then the probability of not being a fan is 1 - .72, or .28.
Events A and B are independent if the equation P(A∩B) = P(A) and P(B) holds true. You can use the equation to check if events are independent by multiplying the probabilities of the two events together to see if they equal the probability of them both happening together.
Teaching and Learning Resources:
- Manipulative (dice, coins, spinners, playing cards, counters, digit cards),
- Simple Probability Mazes (Printable & Digital),
- Worksheets
- Task Cards
- YEAR TWO
Assessment (1.4.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.