Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate probability for combined events.
• Analyse probability statements about everyday risks.
• Apply probability concepts to make informed decisions.
• Differentiate between independent and dependent events.
• Use tree diagrams to represent and calculate probabilities.

Lesson summary

Probability is a fundamental concept in mathematics that helps us understand the likelihood of events occurring. This week, we delve into combined events and how probability influences our everyday decision-making, particularly concerning risks. This knowledge is crucial for South African learners because it empowers you to make informed choices in various aspects of life, from understanding insurance policies and lottery odds to assessing health risks and evaluating the credibility of information. Being able to critically analyze probabilities allows you to navigate a world saturated with data and make more rational decisions.

Lesson notes

2.1 Basic Probability Review Probability: A measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1 (inclusive), where 0 means the event is impossible and 1 means the event is certain.

Event: A specific outcome or set of outcomes.

Sample Space: The set of all possible outcomes of an experiment or situation.

Calculating Probability: Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes) 2.2 Combined Events Combined events involve two or more events occurring. There are several types of combined events, each requiring a different approach to calculating probability. 2.2.1 Mutually Exclusive Events: Definition: Two events are mutually exclusive if they cannot occur at the same time. In other words, they have no outcomes in common.

Example: Rolling a 1 or a 2 on a single die in one roll. You cannot roll a 1 and a 2 simultaneously.

Formula: If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)

Example: A bag contains 3 red balls, 2 blue balls, and 5 green balls. What is the probability of drawing a red ball or a blue ball?

Solution: P(Red) = 3/10, P(Blue) = 2/

1

0. Since drawing a red ball and drawing a blue ball are mutually exclusive, P(Red or Blue) = P(Red) + P(Blue) = 3/10 + 2/10 = 5/10 = 1/2. 2.2.2 Independent Events: Definition: Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.

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

Formula: If A and B are independent events, then P(A and B) = P(A) P(B)

Example: A coin is flipped and a die is rolled. What is the probability of getting heads on the coin and a 6 on the die?

Solution: P(Heads) = 1/2, P(6) = 1/

6. Since the events are independent, P(Heads and 6) = P(Heads) P(6) = (1/2) (1/6) = 1/12. 2.2.3 Dependent Events (Conditional Probability): Definition: Two events are dependent if the occurrence of one event does affect the probability of the other event occurring. Often, these are described using conditional probability.

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

Formula: P(B|A) = P(A and B) / P(A), where P(B|A) is the probability of event B occurring given that event A has already occurred. This is read as "the probability of B given A." Rearranging the formula: P(A and B) = P(A) P(B|A)

Example: A bag contains 5 apples and 3 oranges. Two fruits are drawn without replacement. What is the probability that both fruits are apples?

Solution: Let A be the event that the first fruit is an apple, and B be the event that the second fruit is an apple. P(A) = 5/

8. If the first fruit is an apple, there are now 4 apples and 3 oranges left, so P(B|A) = 4/

7. P(A and B) = P(A) P(B|A) = (5/8) (4/7) = 20/56 = 5/14. 2.3 Venn Diagrams and Tree Diagrams These are visual tools that can help understand and calculate probabilities.

Venn Diagrams: Useful for representing the relationships between sets and events, especially for mutually exclusive and overlapping events. Overlapping sections show the intersection (AND) of events. The total area within the circles represents the union (OR) of events.

Tree Diagrams: Useful for visualizing sequential events, where the outcome of one event affects the probability of subsequent events (dependent events). Each branch represents a possible outcome, and the probabilities are written along the branches. Multiply the probabilities along the branches to find the probability of a specific sequence of events. 2.4 Everyday Risk Probability plays a vital role in understanding and managing risk in our daily lives.

Examples include: Health Risks: Understanding the probability of developing certain diseases based on lifestyle choices or family history. (e.g., risk of lung cancer for smokers)

Financial Risks: Assessing the probability of investment losses or gains, understanding insurance policies, and making informed financial decisions. (e.g., risk of default on a loan)

Safety Risks: Evaluating the probability of accidents in different situations, such as driving, crossing the street, or participating in sports. (e.g., risk of a car accident while texting and driving)

Lottery/Gambling: Understanding the extremely low probability of winning large jackpots. This is a common example, particularly in South Africa with the popularity of the national lottery. Many people spend a significant portion of their income on the lottery without truly understanding the odds. Guided Practice (With Solutions)

Question 1: A survey of 100 students at a school in Soweto revealed that 60 students like soccer, 40 like netball, and 20 like both soccer and netball. a) Draw a Venn diagram to represent this information. b) What is the probability that a randomly selected student likes soccer or netball?