Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate the probability of combined events.
• Apply probability concepts to analyse everyday risks.
• Distinguish between subjective and objective probabilities.
• Use probability to make informed decisions about risk.
• Interpret probability statements from media sources.

Lesson summary

This week, we delve into the fascinating world of probability, specifically focusing on combined events and how probability relates to everyday risks. Understanding probability isn't just about academic exercises; it's a crucial life skill. In South Africa, where we face various socio-economic challenges, being able to assess risk and make informed decisions based on probabilities is essential for financial planning, health choices, and overall well-being. For instance, understanding the probability of winning the Lotto, the likelihood of defaulting on a loan, or the risk of contracting a disease can significantly impact our choices and outcomes.

Lesson notes

2.1 Basic Probability Review Before we delve into combined events, let's recap the fundamentals of probability: Probability: The chance of a specific event occurring. It's expressed as a number between 0 and 1, where 0 means impossible, and 1 means certain.

Event: A specific outcome or set of outcomes.

Sample Space: The set of all possible outcomes.

Calculating Probability: Probability (Event) = (Number of favourable outcomes) / (Total number of possible outcomes)

Example: If you flip a fair coin, the sample space is {Heads, Tails}. The probability of getting Heads is 1/2 or 0.5. 2.2 Combined Events Combined events involve two or more events happening together or in sequence.

Two key concepts are: 2.2.1 Mutually Exclusive Events: Definition: Two events are mutually exclusive if they cannot occur at the same time. The occurrence of one event excludes the possibility of the other occurring.

Example: Tossing a coin – you can get either heads or tails, but not both simultaneously.

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

Example: A bag contains 5 red balls and 3 blue balls. What is the probability of drawing either a red or a blue ball? P(Red) = 5/8 P(Blue) = 3/8 P(Red or Blue) = P(Red) + P(Blue) = 5/8 + 3/8 = 8/8 = 1 2.2.2 Independent Events: Definition: Two events are independent if the outcome of one event does not affect the outcome of the other event.

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

Probability Rule: If events A and B are independent, 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 4 on the die? P(Heads) = 1/2 P(4) = 1/6 P(Heads and 4) = P(Heads) P(4) = (1/2) (1/6) = 1/12 2.3 Dependent Events (Brief Introduction) While not the main focus, it's important to acknowledge dependent events. These are events where the outcome of one event does affect the outcome of the other. The probability rule is slightly more complex and involves conditional probability, P(A and B) = P(A) * P(B|A), where P(B|A) is the probability of B given that A has already occurred. A common example is drawing cards without replacement. 2.4 Everyday Risk and Probability Probability is used to assess risk in various aspects of life.

Financial Decisions: Understanding interest rates, investment returns, and the risk of default helps individuals make sound financial decisions.

Health: Understanding the probability of contracting a disease or the effectiveness of a treatment helps individuals make informed health choices.

Transportation: Understanding accident rates and the probability of vehicle malfunctions helps individuals make safer transportation decisions. 2.5 Subjective vs. Objective Probability Objective Probability: Based on empirical data and mathematical calculations (e.g., the probability of rolling a 6 on a fair die).

Subjective Probability: Based on personal beliefs, experience, or intuition. It's an estimate of the likelihood of an event occurring (e.g., the probability of a certain political party winning an election). Subjective probabilities are prone to bias and should be treated with caution.

Example: The probability of a car accident is objectively determined using accident statistics. The probability of a specific business succeeding is a subjective assessment based on market research and expertise. Guided Practice (With Solutions)

Question 1: A survey shows that 60% of South African households have a television and 40% have a computer. If 25% have both, what is the probability that a household has either a television or a computer?

Solution: Let T be the event "has a television" and C be the event "has a computer." P(T) = 0.6 P(C) = 0.4 P(T and C) = 0.25 P(T or C) = P(T) + P(C) - P(T and C) = 0.6 + 0.4 - 0.25 = 0.75 Therefore, the probability that a household has either a television or a computer is 0.75 or 75%. The key here is to remember to subtract the intersection to avoid double counting.

Question 2: A bag contains 7 red marbles and 3 green marbles. You draw one marble, replace it, and then draw another. What is the probability of drawing a red marble followed by a green marble?

Solution: Since the marble is replaced, the events are independent. P(Red) = 7/10 P(Green) = 3/10 P(Red and Green) = P(Red) P(Green) = (7/10) (3/10) = 21/100 = 0.21 The probability of drawing a red marble followed by a green marble is 0.21 or 21%. Replacement ensures the probabilities remain constant between draws.

Question 3: In a certain South African city, 15% of residents have private medical insurance, and 85% use public healthcare. What is the probability that a randomly selected resident uses public healthcare or has private medical insurance?

Solution: Since a resident will either have private medical insurance, or use public healthcare (we assume that the intersection here is very small, e.g.