Lesson Notes By Weeks and Term v5 - Grade 12
Probability: combined events and everyday risk – Week 7 focus
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Probability: combined events and everyday risk – Week 7 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematical Literacy
Class: Grade 12
Term: 3rd Term
Week: 7
Theme: General lesson support
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Probability is a fundamental concept that governs many aspects of our lives, from weather forecasting to financial investments. In this week's focus, we'll delve into the fascinating world of combined events and everyday risk. Understanding how probabilities interact when multiple events occur is crucial for making informed decisions, evaluating risks, and interpreting statistical information in various real-world scenarios, particularly those relevant to the South African context. For example, understanding the probability of defaulting on a loan, the chances of winning the lottery, or the risk of contracting a disease are all examples where understanding probability is essential.
2.1 Basic Probability Review: Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Formula: 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 simultaneously or sequentially.
Independent Events: Two events are independent if the outcome of one does not affect the outcome of the other.
Multiplication Rule (Independent Events): 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 on die) = 1/6 P(Heads and 4) = (1/2) (1/6) = 1/12 Dependent Events: Two events are dependent if the outcome of one event affects the outcome of the other.
Multiplication Rule (Dependent Events): P(A and B) = P(A) P(B|A), where P(B|A) is the probability of event B occurring given that event A has already occurred (conditional probability).
Example:* A bag contains 5 red balls and 3 blue balls. Two balls are drawn without replacement. What is the probability that both balls are red? P(First ball red) = 5/8 P(Second ball red | First ball red) = 4/7 (since one red ball has been removed) P(Both balls red) = (5/8) (4/7) = 20/56 = 5/14 Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time.
Addition Rule (Mutually Exclusive Events): P(A or B) = P(A) + P(B)
Example:* A die is rolled. What is the probability of rolling a 2 or a 5? P(2) = 1/6 P(5) = 1/6 P(2 or 5) = (1/6) + (1/6) = 2/6 = 1/3 Non-Mutually Exclusive Events: Two events are non-mutually exclusive if they can occur at the same time. Addition Rule (Non-Mutually Exclusive Events): P(A or B) = P(A) + P(B) - P(A and B)
Example:* In a class, 60% of the students like soccer, 40% like rugby, and 20% like both. What is the probability that a student likes soccer or rugby? P(Soccer) = 0.6 P(Rugby) = 0.4 P(Soccer and Rugby) = 0.2 P(Soccer or Rugby) = 0.6 + 0.4 - 0.2 = 0.8 2.3 Everyday Risk: Everyday risks involve probabilities related to events that occur in our daily lives.
These can include: Insurance: Calculating premiums based on the probability of claims.
Example:* An insurance company assesses the probability of a car accident for a young driver at 0.1 per year. This probability, combined with the potential cost of an accident, influences the insurance premium.
Health: Assessing the likelihood of contracting diseases.
Example:* Health officials might state that the probability of contracting a specific strain of flu during flu season is 0.
2. This informs public health campaigns.
Finance: Evaluating the probability of investment returns or loan defaults.
Example:* A bank determines the probability of a loan defaulting at 0.
0
5. This affects the interest rate they charge.
Gambling/Lotteries: Calculating the probability of winning.
Example:* We can use probability to demonstrate the extremely low chances of winning the Lotto or Powerball in South Africa, reinforcing the fact that these are games of chance with a high probability of losing.
Road Safety: Analyzing the probability of road accidents based on factors like driver behavior, road conditions and vehicle maintenance. 2.4 Conditional Probability: Conditional probability deals with the probability of an event occurring given that another event has already occurred. This is particularly important in assessing risk because knowing that one event has already taken place can significantly change the probability of another.
Formula: P(A|B) = P(A and B) / P(B)
Example:* A clinic tests patients for HIV. The test is 99% accurate. If a person tests positive, what is the probability that they actually have HIV, given that the prevalence of HIV in the population is 10%? Let A = Has HIV, B = Tests Positive P(A) = 0.10 P(B|A) = 0.99 (Probability of testing positive given they have HIV) P(B|A') = 0.01 (Probability of testing positive given they DON'T have HIV – False positive) P(A') = 0.90 P(B) = P(B|A)P(A) + P(B|A')P(A') = (0.99)(0.1) + (0.01)(0.9) = 0.099 + 0.009 = 0.108 P(A|B) = P(A and B) / P(B) = P(B|A)P(A)/P(B) = (0.99 0.1)/0.108 = 0.099/0.108 ≈ 0.917, or 91.7% Even with a highly accurate test, a positive result does not guarantee that you have the disease, especially if the prevalence is low. This is very relevant to understanding medical testing in South Africa. Guided Practice (With Solutions)
Question 1: A survey in a South African township found that 70% of households have a television, and 40% have a refrigerator. 25% of households have both. What is the probability that a randomly selected household has either a television or a refrigerator (or both)?