Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate probability for 'AND' events.
• Calculate probability for 'OR' events.
• Use Tree Diagrams for sequential events.
• Calculate conditional probabilities.
• Apply probability to assess everyday risks.

Lesson summary

Probability is the mathematical measure of the likelihood that an event will occur. Understanding probability, especially combined events and risk, is crucial for making informed decisions in everyday life. From understanding the chances of winning the Lotto to assessing the risks associated with taking out a loan, probability skills are essential for navigating the world around us. This week, we will focus on combined events (events that involve two or more outcomes) and how to calculate the probability of these events occurring, as well as how this relates to the assessment of everyday risk in South Africa.

Lesson notes

2.1 Combined Events: "AND" (Intersection) When we want to know the probability of two events A AND B both happening, we are looking for the intersection of the two events. This means we want to know the probability of A and B occurring simultaneously.

Formula (for any events A and B): P(A and B) = P(A) P(B|A) where P(B|A) is the probability of B given that A has already occurred (conditional probability). If A and B are independent, then P(B|A) = P(B), so P(A and B) = P(A) * P(B).

Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin twice – the outcome of the first flip does not influence the outcome of the second flip.

Dependent Events: Two events are dependent if the occurrence of one event does affect the probability of the other. For example, drawing two cards from a deck without replacement – the outcome of the first draw changes the probabilities for the second draw.

Example 1: Independent Events A local spaza shop owner, Thando, estimates that the probability of a customer buying bread is 0.6 and the probability of a customer buying milk is 0.

4. Assuming these events are independent (buying bread doesn't affect the likelihood of buying milk), what is the probability that a customer buys both bread and milk?

Solution: Since the events are independent: P(Bread and Milk) = P(Bread) P(Milk) = 0.6 0.4 = 0.24 Therefore, there is a 24% chance that a customer buys both bread and milk.

Example 2: Dependent Events In a class of 30 learners, 12 are studying Accounting, 15 are studying Business Studies, and 5 are studying both. What is the probability that a randomly selected learner is studying Accounting given that they are studying Business Studies?

Solution: This is a conditional probability question, P(Accounting | Business Studies). P(Accounting | Business Studies) = P(Accounting and Business Studies) / P(Business Studies) P(Accounting and Business Studies) = 5/30 P(Business Studies) = 15/30 P(Accounting | Business Studies) = (5/30) / (15/30) = 5/15 = 1/3 = 0.333 or 33.3% 2.2 Combined Events: "OR" (Union) When we want to know the probability of event A OR event B happening, we are looking for the union of the two events. This includes cases where A happens, B happens, or both happen.

Formula: P(A or B) = P(A) + P(B) - P(A and B) The reason we subtract P(A and B) is to avoid double-counting the outcomes that are in both A and

B. Mutually Exclusive Events: If events A and B are mutually exclusive (they cannot happen at the same time), then P(A and B) =

0. In this case, the formula simplifies to: P(A or B) = P(A) + P(B). For example, flipping a coin cannot land on both heads and tails at the same time.

Example 3: "OR" Events In a survey of 100 people in a township, 60 own a cellphone, 40 own a TV, and 20 own both. What is the probability that a randomly selected person owns a cellphone or a TV?

Solution: P(Cellphone or TV) = P(Cellphone) + P(TV) - P(Cellphone and TV) P(Cellphone or TV) = (60/100) + (40/100) - (20/100) = 80/100 = 0.8 Therefore, there is an 80% chance that a randomly selected person owns a cellphone or a TV. 2.3 Tree Diagrams Tree diagrams are useful for visualizing and calculating probabilities in sequential events. Each branch represents a possible outcome, and the probabilities are written along the branches. To find the probability of a sequence of events, you multiply the probabilities along the corresponding branches.

Example 4: Tree Diagram A student, Ayanda, takes two buses to school. The probability that the first bus is on time is 0.

8. If the first bus is on time, the probability that the second bus is on time is 0.

9. If the first bus is late, the probability that the second bus is on time is 0.

6. What is the probability that Ayanda arrives at school on time (both buses are on time)?

Solution: Step 1: Draw the tree diagram.

First Branch: Bus 1 on time (0.8) or late (0.2) Second Branch (from each branch of the first): Bus 2 on time or late Step 2: Calculate the probability of both buses being on time. This is the path "Bus 1 on time" AND "Bus 2 on time". Probability = 0.8 * 0.9 = 0.72 Therefore, there is a 72% chance that Ayanda arrives at school on time. 2.4 Everyday Risk Probability helps us quantify risk. Risk is the possibility of suffering harm or loss. Understanding probabilities allows us to assess the likelihood of negative outcomes and make more informed decisions.

Financial Risk: Probability of losing money on an investment, not being able to repay a loan, etc.

Health Risk: Probability of contracting a disease, experiencing side effects from medication, etc.

Crime Risk: Probability of being a victim of theft, assault, etc.

Example 5: Risk Assessment A small business owner is considering taking out a loan to expand their business. The bank estimates that there is a 20% chance that the business will fail if they take out the loan.