Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define key probability terms.
• Calculate the probability of simple events.
• Express probability as a fraction, decimal, and percentage.
• Construct sample spaces using lists, tables, and tree diagrams.
• Understand impossible, certain, and likely events on a probability scale.

Lesson summary

Probability is a fundamental concept in Mathematical Literacy that deals with the likelihood of an event occurring. Understanding probability helps us make informed decisions and predictions in situations involving uncertainty. In South Africa, probability concepts are used daily, from understanding weather forecasts affecting agricultural planning to assessing the odds of winning the Lotto or Powerball, which impacts household budgeting. Moreover, probability plays a crucial role in risk assessment in businesses, insurance, and even understanding crime statistics.

Lesson notes

2. 1.

Basic Terminology Experiment: An activity or process that results in an outcome.

Examples: Tossing a coin, rolling a die, drawing a card from a deck, selecting a name from a hat. In a South African context, an experiment could be drawing a winning ticket from a raffle at a school fundraiser.

Outcome: A possible result of an experiment.

Examples: When tossing a coin, the possible outcomes are "Heads" or "Tails." When rolling a die, the possible outcomes are 1, 2, 3, 4, 5, or

6. Sample Space: The set of all possible outcomes of an experiment. It is often denoted by the symbol

S. Example: For a coin toss, S = {Heads, Tails}. For rolling a die, S = {1, 2, 3, 4, 5, 6}. For choosing a coloured ball from a bag containing red, blue, and green balls, S={Red, Blue, Green}.

Event: A specific outcome or set of outcomes of an experiment. It is a subset of the sample space.

Examples: Rolling an even number on a die (Event = {2, 4, 6}), drawing a red card from a deck of cards (Event = {All red cards}), getting at least one head when tossing two coins (Event = {HH, HT, TH}).

Probability: A measure of how likely an event is to occur. It is expressed as a number between 0 and 1 (inclusive), where 0 indicates impossibility and 1 indicates certainty. Probability (P) of an Event (E) = Number of favorable outcomes / Total number of possible outcomes P(E) = n(E) / n(S), where n(E) is the number of outcomes in event E and n(S) is the number of outcomes in the sample space S. 2.2 Calculating Probability To calculate the probability of an event, we need to determine the number of favorable outcomes and the total number of possible outcomes. This is crucial for making informed decisions in various scenarios.

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

Experiment: Rolling a die.

Sample Space: S = {1, 2, 3, 4, 5, 6}. Total number of possible outcomes, n(S) =

6. Event: Rolling a

4. Event E = {4}. Number of favorable outcomes, n(E) =

1. Probability: P(Rolling a 4) = n(E) / n(S) = 1/

6. Therefore, the probability of rolling a 4 is 1/6, which can be converted to approximately 0.167 or 16.7%.

Example 2: Tossing a Coin What is the probability of getting heads when tossing a fair coin?

Experiment: Tossing a coin.

Sample Space: S = {Heads, Tails}. Total number of possible outcomes, n(S) =

2. Event: Getting Heads. Event E = {Heads}. Number of favorable outcomes, n(E) =

1. Probability: P(Getting Heads) = n(E) / n(S) = 1/

2. Therefore, the probability of getting heads is 1/2, which is equal to 0.5 or 50%.

Example 3: Drawing a Ball from a Bag A bag contains 3 red balls, 2 blue balls, and 5 green balls. What is the probability of drawing a blue ball at random?

Experiment: Drawing a ball from the bag.

Sample Space: S = {3 Red, 2 Blue, 5 Green}. Total number of possible outcomes, n(S) = 3 + 2 + 5 =

1

0. Event: Drawing a Blue ball. Event E = {2 Blue}. Number of favorable outcomes, n(E) =

2. Probability: P(Drawing a Blue ball) = n(E) / n(S) = 2/10 = 1/

5. Therefore, the probability of drawing a blue ball is 1/5, which is equal to 0.2 or 20%.

Example 4: Raffle Ticket A class of 30 students holds a raffle. You buy 3 tickets. What is the probability that you will win?

Experiment: Drawing a winning raffle ticket.

Sample Space: S = {30 tickets}. Total number of possible outcomes, n(S) = 30 Event: You win. Event E = {3 tickets}. Number of favorable outcomes, n(E) =

3. Probability: P(Winning) = n(E) / n(S) = 3/30 = 1/

1

0. Therefore, the probability of you winning is 1/10, or 10%. 2.3 Expressing Probability Probability can be expressed as a fraction, a decimal, or a percentage: Fraction: Represents the ratio of favorable outcomes to total outcomes. (e.g., 1/2, 1/6, 2/10).

Decimal: Obtained by dividing the numerator of the fraction by the denominator. (e.g., 0.5, 0.167, 0.2).

Percentage: Obtained by multiplying the decimal by 100. (e.g., 50%, 16.7%, 20%). All three representations are equivalent and convey the same information about the likelihood of an event. 2.4 Sample Space Representation Lists: Simply listing all possible outcomes. (e.g., Coin Toss: {Heads, Tails})

Tables: Useful for visualizing outcomes when there are two or more experiments. For example, tossing two coins: | Coin 1 | Coin 2 | Outcome | | :----- | :----- | :------ | | Heads | Heads | HH | | Heads | Tails | HT | | Tails | Heads | TH | | Tails | Tails | TT | Tree Diagrams: Visual representations that branch out to show possible outcomes. Useful for sequential experiments. For example, tossing a coin twice: ``` Start / \ Heads (1/2) Tails (1/2) / \ / \ Heads(1/2) Tails(1/2) Heads(1/2) Tails(1/2) HH HT TH TT ``` 2.5 Certain, Impossible, and Likely Events Certain Event: An event that is guaranteed to happen. Its probability is 1 (or 100%).

Example: The sun will rise tomorrow.

Impossible Event: An event that cannot happen. Its probability is 0 (or 0%).

Example: Rolling a 7 on a standard six-sided die.