Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate probabilities of simple and compound events.
• Interpret probabilities as measures of risk.
• Predict outcomes in real-life scenarios.
• Critically evaluate probability claims in media.
• Analyse risk in situations like road safety or finances.

Lesson summary

Probability is the study of chance and the likelihood of events occurring. In Mathematical Literacy, we focus on using probability to make informed decisions about risk and to predict potential outcomes in real-life situations. Understanding probability is crucial for South African learners because it equips you with the skills to assess risks associated with various aspects of your lives, such as financial investments, health choices, and even career paths. For example, understanding the probability of winning the lottery can help you make an informed decision about whether to participate.

Lesson notes

2.1 Basic Probability: Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain. The probability of an event A is written as P(A). The basic formula for calculating probability is: P(A) = (Number of favourable outcomes) / (Total number of possible outcomes)

Example 1: A standard six-sided die is rolled. What is the probability of rolling a 4?

Favourable outcome: rolling a 4 (1 outcome)

Total possible outcomes: 1, 2, 3, 4, 5, 6 (6 outcomes) P(rolling a 4) = 1/6 2.2 Types of Events: Simple Event: An event with a single outcome (e.g., rolling a 4 on a die).

Compound Event: An event with two or more possible outcomes (e.g., rolling an even number on a die).

Independent Events: Events where the outcome of one event does not affect the outcome of the other event (e.g., flipping a coin and then rolling a die).

Dependent Events: Events where the outcome of one event does affect the outcome of the other event (e.g., drawing a card from a deck without replacing it).

Mutually Exclusive Events: Events that cannot occur at the same time (e.g., flipping a coin and getting both heads and tails simultaneously). 2.3 Calculating Probabilities of Compound Events: Independent Events: P(A and B) = P(A)

P(B)

Mutually Exclusive Events: P(A or B) = P(A) + P(B)

Non-Mutually Exclusive Events: P(A or B) = P(A) + P(B) - P(A and B)

Example 2: Independent Events A coin is flipped, and a die is rolled. What is the probability of getting tails on the coin and a 3 on the die? P(tails) = 1/2 P(rolling a 3) = 1/6 P(tails and rolling a 3) = (1/2) * (1/6) = 1/12 Example 3: Mutually Exclusive Events In a bag, there are 5 red balls and 3 blue balls. What is the probability of drawing a red ball or a blue ball? P(red ball) = 5/8 P(blue ball) = 3/8 P(red ball or blue ball) = 5/8 + 3/8 = 8/8 = 1 (This is certain since there are only red and blue balls)

Example 4: Non-Mutually Exclusive Events In a class of 30 students, 15 take Mathematics, 12 take Physics, and 5 take both. What is the probability that a student selected at random takes Mathematics or Physics? P(Mathematics) = 15/30 P(Physics) = 12/30 P(Mathematics and Physics) = 5/30 P(Mathematics or Physics) = (15/30) + (12/30) - (5/30) = 22/30 = 11/15 2.4 Tree Diagrams: Tree diagrams are useful for visualizing and calculating probabilities of sequential events, particularly when dealing with dependent events. Each branch represents a possible outcome, and the probabilities are written along the branches.

Example 5: Tree Diagram A bag contains 2 red balls and 3 blue balls. A ball is drawn, and not replaced, and then another ball is drawn. What is the probability of drawing two red balls?

First draw: P(Red) = 2/5 P(Blue) = 3/5 Second draw (dependent on the first): If the first ball was red: P(Red) = 1/4 (since one red ball has been removed) P(Blue) = 3/4 If the first ball was blue: P(Red) = 2/4 P(Blue) = 2/4 To find the probability of drawing two red balls: P(Red then Red) = (2/5) * (1/4) = 2/20 = 1/10 2.5 Two-Way Tables: Two-way tables organize data based on two categorical variables. They are helpful for calculating conditional probabilities.

Example 6: Two-Way Table A survey was conducted among 100 adults in a South African township to determine their employment status and level of education.

The results are shown below: | | Employed | Unemployed | Total | |-----------------|----------|------------|-------| | Matric | 35 | 15 | 50 | | No Matric | 20 | 30 | 50 | | Total | 55 | 45 | 100 | What is the probability that a randomly selected adult is employed given that they have a Matric certificate? P(Employed | Matric) = (Number of Employed with Matric) / (Total with Matric) = 35/50 = 7/10 2.6 Predicting Outcomes and Risk Probability allows us to make predictions about the likelihood of future events and to assess the risk associated with different choices. Understanding the probability of a specific outcome can help individuals and businesses make more informed decisions. For example, insurance companies use probability to estimate the risk of insuring individuals or property.

Example 7: Insurance Risk An insurance company estimates that the probability of a car being involved in an accident in a year is 0.

0

5. If the company insures 1000 cars, how many accidents can they expect in a year? Expected number of accidents = (Probability of an accident) (Number of cars) = 0.05 1000 = 50 accidents Guided Practice (With Solutions)

Question 1: A bag contains 4 green marbles and 6 yellow marbles. What is the probability of randomly selecting a green marble from the bag?

Solution: Number of favourable outcomes (green marbles): 4 Total number of possible outcomes (total marbles): 4 + 6 = 10 P(Green) = 4/10 = 2/5 Question 2: Two dice are rolled. What is the probability that the sum of the numbers rolled is 7?