Lesson Notes By Weeks and Term v5 - Grade 10
Probability – Week 2 focus
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Probability – Week 2 focus
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Subject: Mathematics
Class: Grade 10
Term: Term 4
Week: 2
Theme: General lesson support
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This week, we delve deeper into the fascinating world of probability, building on the basic concepts you learned last week. Understanding probability is crucial for making informed decisions in everyday life. From predicting the likelihood of rain so you know whether to pack an umbrella, to understanding the odds in the lottery or a sporting event, probability provides a framework for dealing with uncertainty. In South Africa, understanding probability is vital for interpreting statistical data related to topics like crime rates, unemployment figures, or the spread of diseases, allowing you to critically evaluate information and participate more effectively in society.
Compound Events A compound event is an event that consists of two or more simple events occurring together. We often use tree diagrams and two-way tables to visualize and calculate the probabilities of compound events.
Tree Diagrams: A tree diagram is a visual tool used to illustrate all possible outcomes of a sequence of events. Each branch represents a possible outcome, and the probability of each outcome is written along the branch. The probabilities along each set of branches from a single point must add up to
1. Two-Way Tables: A two-way table (also known as a contingency table) is a table that displays the frequencies or probabilities of two categorical variables. It helps to organize and analyze data involving two or more attributes.
Example 1: Coin Toss and Dice Roll (Tree Diagram) Let's say you flip a fair coin and then roll a fair six-sided die. What is the probability of getting heads on the coin and an even number on the die?
Step 1: Draw the tree diagram.
First branch: Coin flip (Heads or Tails), each with probability 1/
2. Second branch from Heads: Dice roll (1, 2, 3, 4, 5, 6), each with probability 1/
6. Second branch from Tails: Dice roll (1, 2, 3, 4, 5, 6), each with probability 1/
6. Step 2: Identify the desired outcome. We want Heads and an even number (2, 4, or 6).
Step 3: Calculate the probability. The probability of Heads is 1/
2. The probability of an even number is 3/6 = 1/
2. Since these are independent events (the coin flip doesn't affect the die roll), we multiply the probabilities: (1/2) * (1/2) = 1/4 Therefore, the probability of getting heads and an even number is 1/
4. Example 2: Survey on Favourite Sport (Two-Way Table) A survey was conducted among 200 learners at a school in Soweto to determine their favorite sport. The results are shown in the following two-way table: | Sport | Boys | Girls | Total | |------------|------|-------|-------| | Soccer | 60 | 30 | 90 | | Netball | 10 | 50 | 60 | | Cricket | 30 | 20 | 50 | | Total | 100 | 100 | 200 | What is the probability that a randomly selected learner is a girl who prefers netball?
Step 1: Identify the relevant cell. Look for the intersection of "Girls" and "Netball" in the table. The value is
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0. Step 2: Calculate the probability. Divide the number of girls who prefer netball by the total number of learners: 50 / 200 = 1/4 Therefore, the probability that a randomly selected learner is a girl who prefers netball is 1/
4. Addition Rule The addition rule is used to find the probability of either event A OR event B occurring.
Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. The probability of event A or event B occurring is: P(A or B) = P(A) + P(B)
Non-Mutually Exclusive Events: Two events are non-mutually exclusive if they can occur at the same time. The probability of event A or event B occurring is: P(A or B) = P(A) + P(B) - P(A and B)
Example 3: Drawing a Card (Mutually Exclusive) What is the probability of drawing a heart or a spade from a standard deck of 52 cards?
Step 1: Identify the events. Event A is drawing a heart, and event B is drawing a spade. These are mutually exclusive because you can't draw a card that is both a heart and a spade.
Step 2: Calculate the probabilities. P(Heart) = 13/52 = 1/4 P(Spade) = 13/52 = 1/4 Step 3: Apply the addition rule. P(Heart or Spade) = P(Heart) + P(Spade) = (1/4) + (1/4) = 1/2 Therefore, the probability of drawing a heart or a spade is 1/
2. Example 4: Taking Maths or Science (Non-Mutually Exclusive) In a class of 30 learners, 15 take Mathematics, 12 take Science, and 5 take both Mathematics and Science. What is the probability that a randomly selected learner takes either Mathematics or Science?
Step 1: Identify the events. Event A is taking Mathematics, and event B is taking Science. These are not mutually exclusive because some learners take both.
Step 2: Calculate the probabilities. P(Maths) = 15/30 = 1/2 P(Science) = 12/30 = 2/5 P(Maths and Science) = 5/30 = 1/6 Step 3: Apply the addition rule. P(Maths or Science) = P(Maths) + P(Science) - P(Maths and Science) = (1/2) + (2/5) - (1/6) = (15 + 12 - 5) / 30 = 22/30 = 11/15 Therefore, the probability that a randomly selected learner takes either Mathematics or Science is 11/
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5. Multiplication Rule The multiplication rule is used to find the probability of both event A AND event B occurring.
Independent Events: Two events are independent if the outcome of one event does not affect the outcome of the other. The probability of event A and event B occurring is: P(A and B) = P(A) * P(B)
Dependent Events: Two events are dependent if the outcome of one event does affect the outcome of the other. The probability of event A and event B occurring is: P(A and B) = P(A) * P(B|A) where P(B|A) is the conditional probability of event B occurring given that event A has already occurred.