Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate probability of combined events using tables and diagrams.
• Apply probability to analyse everyday risks (health, safety, finance).
• Distinguish between independent and dependent events.
• Critically evaluate probability claims in the media.
• Use probability to make informed decisions.

Lesson summary

Probability is not just a theoretical concept; it's a powerful tool that helps us understand and navigate the world around us. This week, we'll be focusing on combined events and understanding the risks we face daily in South Africa. We'll explore how probabilities of multiple events occurring together can be calculated and interpreted, and how this knowledge can inform our decision-making. This is particularly relevant in a country where resources are often limited, and understanding risks is crucial for making informed choices about health, finances, and safety.

Lesson notes

2.1 Basic Probability Review Before we dive into combined events, let's recap the basics. 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. Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Example: If you flip a fair coin, the probability of getting heads is 1/2 or 0.5. 2.2 Combined Events Combined events involve two or more events happening together. The key is to understand how these events relate to each other – are they independent, dependent, or mutually exclusive? 2.2.1 Mutually Exclusive Events: Mutually exclusive events are events that cannot occur at the same time. If one event happens, the other cannot.

Example: Tossing a coin - you can get heads or tails, but you cannot get both at the same time.

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

Example: A school tuck shop sells cooldrinks: 30% are Coke, 20% are Fanta, and 50% are Sprite. What is the probability of a student buying either Coke or Fanta? P(Coke or Fanta) = P(Coke) + P(Fanta) = 0.30 + 0.20 = 0.50 or 50% 2.2.2 Independent Events: Independent events are events where the outcome of one event does not affect the outcome of the other.

Example: Tossing a coin twice. The result of the first toss doesn't influence the result of the second toss.

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

Example: The probability of rain on Monday is 0.4, and the probability of the Proteas winning their cricket match on Tuesday is 0.

6. Assuming these events are independent, what's the probability of it raining on Monday and the Proteas winning on Tuesday? P(Rain and Proteas Win) = P(Rain) P(Proteas Win) = 0.4 * 0.6 = 0.24 or 24% 2.2.3 Dependent Events: Dependent events are events where the outcome of one event does affect the outcome of the other.

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

Formula: If A and B are dependent, then P(A and B) = P(A) P(B|A), where P(B|A) is the probability of B given that A has already occurred.

Example: A bag contains 5 red marbles and 3 blue marbles. You draw one marble and don't replace it. Then you draw another marble. What's the probability of drawing a red marble followed by another red marble? P(Red first) = 5/8 P(Red second | Red first) = 4/7 (Since one red marble has been removed) P(Red and Red) = (5/8) (4/7) = 20/56 = 5/14 or approximately 35.7% 2.3 Two-Way Tables and Venn Diagrams These tools are useful for organizing and visualizing probabilities related to combined events. Two-Way Table

Example: A survey of 200 students at a Durban high school revealed the following information about their participation in sports and cultural activities: | | Play Sports | Don't Play Sports | Total | |-----------------|-------------|-------------------|-------| | Participate in Cultural Activities | 60 | 40 | 100 | | Don't Participate in Cultural Activities | 70 | 30 | 100 | | Total | 130 | 70 | 200 | What is the probability that a randomly selected student plays sports and participates in cultural activities? P(Sports and Cultural) = 60/200 = 0.3 or 30% What is the probability that a randomly selected student plays sports or participates in cultural activities? P(Sports or Cultural) = P(Sports) + P(Cultural) - P(Sports and Cultural) = 130/200 + 100/200 - 60/200 = 170/200 = 0.85 or 85% Venn Diagram

Example: Suppose we have 100 residents in a township. 60 own a radio, 40 own a television, and 20 own both. Draw a Venn diagram representing this information. What is the probability that a resident owns either a radio or a television?

Solution: Draw two overlapping circles, one for radios (R) and one for televisions (T). The overlapping region represents residents who own both, so write '20' in the intersection. Since 60 own a radio in total, and 20 own both, then 60 - 20 = 40 own only a radio. Write '40' in the R circle only. Similarly, 40 own a television in total, and 20 own both, so 40 - 20 = 20 own only a television. Write '20' in the T circle only. The total number inside both circles is 40 + 20 + 20 =

8

0. That means 100 - 80 = 20 residents own neither. P(Radio or Television) = P(Radio) + P(Television) - P(Radio and Television) = 60/100 + 40/100 - 20/100 = 80/100 = 0.8 or 80% 2.4 Everyday Risk Understanding probability helps us assess and manage risks in everyday life.

Consider these examples: Health: Understanding the probability of contracting HIV/AIDS based on certain behaviors encourages safer choices. Knowledge of the probability of developing certain cancers based on lifestyle choices (smoking, diet) informs preventative measures.

Finance: Evaluating the risk of investments. High-yield investments often come with higher risk. Understanding the probabilities of different returns helps make informed decisions.