Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Calculate the probability of a single event.
• Represent sample spaces using lists, tables, and tree diagrams.
• Understand the difference between experimental and theoretical probability.
• Solve real-life problems involving probability.
• Identify impossible, unlikely, likely, and certain events.

Lesson summary

Data handling and probability are essential mathematical skills that allow us to understand and interpret information, make predictions, and assess risks. In South Africa, these skills are crucial for understanding socio-economic trends, interpreting news reports, making informed financial decisions, and even evaluating the effectiveness of community programs. For example, understanding statistics about unemployment rates, crime statistics, or the spread of diseases allows us to make better decisions and contribute to informed discussions about important issues affecting our country.

Lesson notes

2.1 Understanding Probability Probability is the measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. Probability can be expressed as a fraction, decimal, or percentage.

Formula: Probability of an event (P(event)) = (Number of favourable outcomes) / (Total number of possible outcomes) 2.2 Sample Space The sample space is the set of all possible outcomes of an experiment. For example, if you flip a coin, the sample space is {Heads, Tails}. If you roll a die, the sample space is {1, 2, 3, 4, 5, 6}. 2.3 Representing Sample Spaces Lists: Simply listing all possible outcomes. (e.g., Tossing a coin twice: {HH, HT, TH, TT})

Tables: Useful for representing outcomes of two or more events.

Tree Diagrams: A visual representation of all possible outcomes, especially useful for sequential events. 2.4 Theoretical vs. Experimental Probability Theoretical Probability: The probability of an event based on mathematical calculations, assuming all outcomes are equally likely.

Experimental Probability: The probability of an event based on the results of an experiment. It is calculated by dividing the number of times the event occurred by the total number of trials.

Example 1: Rolling a Die What is the probability of rolling a 4 on a standard six-sided die?

Solution: Number of favourable outcomes (rolling a 4): 1 Total number of possible outcomes (rolling any number from 1 to 6): 6 P(rolling a 4) = 1/6 Example 2: Tossing a Coin What is the probability of getting heads when tossing a fair coin?

Solution: Number of favourable outcomes (getting heads): 1 Total number of possible outcomes (getting heads or tails): 2 P(getting heads) = 1/2 Example 3: Drawing a Card A standard deck of cards has 52 cards, with 13 cards in each suit (hearts, diamonds, clubs, spades). What is the probability of drawing a heart?

Solution: Number of favourable outcomes (drawing a heart): 13 Total number of possible outcomes (drawing any card): 52 P(drawing a heart) = 13/52 = 1/4 Example 4: Tree Diagram (Tossing a Coin Twice) Construct a tree diagram to show the possible outcomes of tossing a coin twice.

Solution: ``` 1st Toss 2nd Toss Outcome / \ / \ H T H T HH, HT / \ / \ H T H T TH, TT ``` The possible outcomes are HH, HT, TH, and T

T. Example 5: Experimental Probability You flip a coin 20 times and get heads 12 times. What is the experimental probability of getting heads?

Solution: Number of times heads occurred: 12 Total number of trials: 20 Experimental P(getting heads) = 12/20 = 3/5 Guided Practice (With Solutions)

Question 1: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of picking a red marble at random?

Solution: Number of red marbles (favourable outcomes): 5 Total number of marbles: 5 + 3 + 2 = 10 P(picking a red marble) = 5/10 = 1/2

Commentary: This question requires applying the basic probability formula. Ensure learners understand how to identify the favourable and total outcomes.

Question 2: A spinner has 4 equal sections coloured yellow, blue, green, and red. What is the probability of landing on blue or green?

Solution: Number of sections that are blue or green (favourable outcomes): 2 Total number of sections: 4 P(landing on blue or green) = 2/4 = 1/2

Commentary: This question introduces the concept of "or," meaning we count both blue and green as favourable outcomes.

Question 3: You roll a six-sided die. What is the probability of rolling an even number?

Solution: Even numbers on a die: 2, 4, 6 (3 favourable outcomes)

Total number of outcomes: 6 P(rolling an even number) = 3/6 = 1/2

Commentary: Reinforces the concept of favorable outcomes and simplifies to lowest terms.

Question 4: What is the probability of choosing a vowel from the word "MATHEMATICS"?

Solution: Vowels in the word "MATHEMATICS": A, E, A, I (4 favourable outcomes)

Total number of letters: 11 P(choosing a vowel) = 4/11

Commentary: This incorporates language skills to identify vowels. Independent Practice (Questions Only) A box contains 7 apples, 5 oranges, and 3 bananas. What is the probability of picking an orange at random? A spinner has 5 equal sections numbered 1 to

5. What is the probability of landing on an odd number? You roll a six-sided die. What is the probability of rolling a number greater than 4? What is the probability of choosing the letter 'S' from the word "SUCCESS"? A bag contains 4 blue beads and 6 yellow beads. If you pick one bead, replace it, and then pick another, what is the probability of picking a blue bead both times? (Hint: Think about the probability of each event happening independently). List all the possible outcomes when flipping a coin and rolling a die simultaneously. Represent it in a table. After surveying 50 students, it was found that 30 students prefer soccer. What is the experimental probability that a student chosen at random prefers soccer?