Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define key probability terms with South African examples.
• Calculate the probability of simple events.
• Understand and calculate the probability of an event not occurring.
• Compare experimental and theoretical probability.
• Explain the difference between independent and dependent events.

Lesson summary

Probability is a crucial concept in Mathematical Literacy as it helps us understand the likelihood of events occurring. This understanding is essential for making informed decisions in various aspects of life, from budgeting and risk assessment to understanding weather forecasts and game strategies. In a South African context, probability can help learners understand issues such as the lottery (Lotto), insurance policies, employment probabilities (likelihood of finding a job), and even the spread of diseases like HIV/AIDS or COVID-

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9. By understanding probability, learners can become more critical and informed citizens.

Lesson notes

2.1 Basic Terminology: Experiment: An activity involving chance that leads to results or outcomes.

Example:* Flipping a coin, rolling a die, drawing a card from a deck.

In a South African context: Picking a number in Lotto, selecting a student randomly from a class, conducting a survey about household income.

Outcome: A possible result of an experiment.

Example:* If the experiment is flipping a coin, the outcomes are "Heads" or "Tails". If the experiment is rolling a die, the outcomes are 1, 2, 3, 4, 5, or

6. In the Lotto example, the outcome is the set of 6 numbers drawn.

Sample Space: The set of all possible outcomes of an experiment. We often denote the sample space by the letter

S. Example:* For flipping a coin, S = {Heads, Tails}. For rolling a die, S = {1, 2, 3, 4, 5, 6}. For Lotto, the sample space is the set of all possible combinations of 6 numbers chosen from a given range (typically 1-52 or 1-49).

Event: A specific outcome or a set of outcomes of an experiment.

Example:* If the experiment is rolling a die, the event could be "rolling an even number." The outcomes that make up this event are {2, 4, 6}.

Another example: Getting at least one head when flipping two coins.

Probability: A measure of how likely an event is to occur. It is a number between 0 and 1 (inclusive), where 0 indicates impossibility and 1 indicates certainty. 2.2 Calculating Probability: The probability of an event occurring is calculated as follows: P(Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

Where: Favorable outcomes are the outcomes that satisfy the event. Total number of possible outcomes is the size of the sample space.

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

Event:* Rolling a 4 Favorable outcomes:* {4} (only one outcome)

Total number of possible outcomes:* {1, 2, 3, 4, 5, 6} (six outcomes) P(rolling a 4) = 1 / 6 Example 2: Drawing a Card What is the probability of drawing a heart from a standard deck of 52 playing cards?

Event:* Drawing a heart Favorable outcomes:* 13 hearts Total number of possible outcomes:* 52 cards P(drawing a heart) = 13 / 52 = 1 / 4 Example 3: South African Context - Unemployment Suppose a town has a total working-age population of 10,000 people. 2,500 of them are unemployed. What is the probability of randomly selecting an unemployed person from the town?

Event:* Selecting an unemployed person Favorable outcomes:* 2,500 unemployed people Total number of possible outcomes:* 10,000 working-age people P(selecting an unemployed person) = 2500 / 10000 = 1 / 4 = 0.25 or 25% 2.3 Complementary Events: The complement of an event A is the event that A does not occur. We denote the complement of A as A'. The probability of the complement of an event is given by: P(A') = 1 - P(A)

Example: If the probability of rain tomorrow is 0.3, what is the probability that it will not rain tomorrow? P(Rain) = 0.3 P(No Rain) = 1 - P(Rain) = 1 - 0.3 = 0.7 2.4 Experimental vs.

Theoretical Probability: Theoretical Probability: The probability calculated using mathematical reasoning and understanding of the situation (as shown in the examples above).

Experimental Probability: The probability calculated based on conducting an experiment multiple times and observing the outcomes.

It is calculated as: Experimental Probability = (Number of times the event occurs) / (Total number of trials)

Example: You flip a coin 20 times and get heads 12 times.

Event:* Getting heads Number of times the event occurs:* 12 Total number of trials:* 20 Experimental Probability of getting heads = 12 / 20 = 3 / 5 = 0.6 Theoretical probability of getting heads = 1 / 2 = 0.5 Notice that experimental probability may not always be equal to theoretical probability, especially with a small number of trials. The more trials you conduct, the closer the experimental probability will likely get to the theoretical probability. 2.5 Independent vs.

Dependent Events: Independent Events: Two events are independent if the outcome of one event does not affect the outcome of the other.

Example:* Flipping a coin twice. The result of the first flip does not influence the result of the second flip.

In a South African context: Winning the lottery one week doesn't increase or decrease your chances of winning the next week.

Dependent Events: Two events are dependent if the outcome of one event does affect the outcome of the other.

Example: Drawing two cards from a deck without replacement. The probability of drawing a particular card on the second draw depends on what card was drawn on the first draw.

In a South African context: if a factory is shortlisting candidates for a job. The fewer people remaining in the shortlist, the higher the chance of being selected. Guided Practice (With Solutions)

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