SHS1 Mathematics · Semester 2 Week 20: Probability/chance

Lesson notes

Learning Objectives

Indicator: 1.4.2.LI.3 - Solve everyday life problems involving the probability of two independent events.

By the end of the lesson, learners can:

Identify two independent events in a real-life problem and state the probability of each event.
Apply the multiplication rule for independent events, P(A and B) = P(A) × P(B), to find the probability that both events occur.
Apply the addition rule for independent events, P(A or B) = P(A) + P(B) - P(A) × P(B), to find the probability that at least one of two events occurs.
Use the complement rule, P(not A) = 1 - P(A), to find the probability that an event does not occur, including cases involving two independent events.
Solve multi-step word problems involving two independent events and express answers as fractions, decimals, percentages or ratios.

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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.3 - Solve everyday life problems involving the probability of two independent events.
Suggested placement: Semester 2, Week 20 (Week 40 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.
Curriculum reference: NaCCA curriculum document, p. 152

Exemplars (from the NaCCA curriculum)

Put learners in convenient groups and offer them appropriate and adequate resources to create and solve some real-life problems.
Example 1: A message is transmitted from Node-A to Node-B through three intermediate nodes. The message will be successfully transmitted only if all the intermediate nodes are working. The probability that an intermediate node will fail is 1%. All nodes are independent of each other. What is the probability that you will not successfully transmit the message?
Solution: We first find the probability that you will successfully transmit the message. For successful transmission, we need all nodes to be working. The probability that a node will not fail is P(Node does not fail) = 1- P(Node fails) = 1 - 0.01 = 0.99. Since all the nodes are independent, the probability that node 1 AND node 2 AND node 3 do not fail = 0.99 × 0.99 × 0.99 = 0.97 Accordingly, P(message is not successful) = 1 - P(message is successful) = 1-0.97 = 0.03 = 3%.
Example 2: You are travelling from location A to location B using a bus and a train. The probability that the bus will get delayed is 10%, and the probability that the train will get delayed is 5%. Both events are independent. Find the probability that:
- You will experience a delay during your travelling.
- You will get on time to location B.
Solution: Let E1 represent the event that the bus gets delayed, and E2 is the event that the train gets delayed.
We will experience a delay if either the bus gets delayed, or the train gets delayed, or both get delayed. So, we are interested in finding the probability P(E1 OR E2) = P(E1 ∪ E2). Using the formulae for the independent events, we can write. P(E1 ∪ E2) = P(E1) + P(E2) - P(E1)P(E2) P(E1 ∪ E2) = 0.1 + 0.05 - (0.1)(0.5) = 0.145 = 14.5%
To get on time, we need neither the bus gets delayed, nor the train gets delayed. Hence, we need to find the probability. P(NOT E1 AND NOT E2) = P(NOT E1 ∩ NOT E2) P(NOT E1) = 1 - P (E1) = 1 - 0.1 = 0.9. P(NOT E2) = 1 - P(E2) = 1 - 0.05 = 0.95.
Since both events are independent P(NOT E1 ∩ NOT E2) = P(NOT E1) x P(NOT E2).
P(NOT E1 ∩ NOT E2) = 0.9 × 0.95 = 0.855 = 85.5%
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.3). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding.