Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate probability of combined events (mutually exclusive & independent).
• Apply probability to analyse everyday risks.
• Use Venn and tree diagrams to represent events.
• Distinguish between theoretical and experimental probability.

Lesson summary

This week, we delve into the fascinating world of probability, specifically focusing on combined events and their connection to everyday risks. Probability isn't just a mathematical concept; it's a tool that helps us understand and make informed decisions about the uncertainties we face daily. From deciding whether to take an umbrella based on the weather forecast to assessing the risk of investing in a particular business, probability plays a crucial role in our lives.

Lesson notes

2. 1.

Basic Probability Refresher: Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1 (inclusive), where: 0 means the event is impossible. 1 means the event is certain. Values between 0 and 1 represent varying degrees of likelihood.

The basic formula for probability is: ``` Probability of an event = (Number of favourable outcomes) / (Total number of possible outcomes) ```

Example: A standard six-sided die is rolled. The probability of rolling a 4 is 1/6 because there's only one "4" and six possible outcomes (1, 2, 3, 4, 5, 6). 2.

2. Combined Events: Combined events involve more than one event occurring.

There are two main types we'll focus on: Mutually Exclusive Events: These events cannot happen at the same time. If one occurs, the other cannot.

Example: Flipping a coin. You can get heads OR tails, but not both simultaneously. Probability of A OR B (mutually exclusive): P(A or B) = P(A) + P(B)

Example: Imagine a bag contains 5 red marbles and 3 blue marbles. What's the probability of picking a red marble OR a blue marble in a single draw? P(Red) = 5/8 P(Blue) = 3/8 P(Red or Blue) = P(Red) + P(Blue) = 5/8 + 3/8 = 8/8 = 1 (It's certain you'll pick either red or blue).

Independent Events: These events do not affect each other. The outcome of one event doesn't influence the outcome of the other.

Example: Flipping a coin twice. The result of the first flip has no impact on the result of the second flip.

Probability of A AND B (independent): P(A and B) = P(A) * P(B)

Example: What's the probability of flipping a coin and getting heads, AND then rolling a die and getting a 6? P(Heads) = 1/2 P(Rolling a 6) = 1/6 P(Heads and 6) = P(Heads) P(Rolling a 6) = (1/2) (1/6) = 1/12 2.

3. Venn Diagrams and Tree Diagrams: These diagrams are visual tools to represent probabilities, especially combined events.

Venn Diagrams: Useful for showing the relationships between sets of events, including overlap (intersection) and mutually exclusive events. The overlapping section represents events that occur simultaneously.

Example: In a class of 30 learners, 15 play soccer, 12 play basketball, and 5 play both. Draw a Venn diagram to represent this. The number of learners who play ONLY soccer would be 15-5 =

1

0. The number who play ONLY basketball would be 12-5 =

7. The number who play neither is 30 - 10 - 7 - 5 =

8. Tree Diagrams: Excellent for visualizing sequential events and calculating probabilities of combined outcomes. Each branch represents a possible outcome, and the probabilities along each branch are multiplied to find the probability of that particular sequence.

Example: A coin is flipped twice. Draw a tree diagram and calculate the probability of getting two heads.

First Flip: Heads (1/2) or Tails (1/2)

Second Flip (after Heads): Heads (1/2) or Tails (1/2)

Second Flip (after Tails): Heads (1/2) or Tails (1/2) The path for two heads is Heads -> Heads, with a probability of (1/2) * (1/2) = 1/4. 2.

4. Everyday Risk and Probability: Many risks we face daily can be analyzed using probability. This involves understanding the likelihood of a negative event occurring and the potential consequences.

Examples: Road Accidents: Factors like speeding, drunk driving, and poor road conditions increase the probability of an accident.

Health Risks: Smoking increases the probability of lung cancer. Unprotected sex increases the probability of STIs.

Financial Risks: Investing in volatile stocks has a higher probability of large gains, but also a higher probability of significant losses.

Example: A news report states that the probability of contracting a certain disease in a specific area is 1 in

5

0

0. What does this mean? It means that for every 500 people in that area, statistically, one person is likely to contract the disease. It doesn't guarantee that someone will get sick, but it highlights the level of risk. 2.

5. Theoretical vs.

Experimental Probability: Theoretical Probability: What we expect to happen based on mathematical calculations. (e.g., the theoretical probability of flipping a fair coin and getting heads is 1/2).

Experimental Probability: What actually happens when we conduct an experiment or observe real-world events. (e.g., flipping a coin 100 times and getting heads 45 times. The experimental probability of getting heads is 45/100). The more times an experiment is repeated (more trials), the closer the experimental probability usually gets to the theoretical probability. This is known as the Law of Large Numbers. Guided Practice (With Solutions)

Question 1: A bag contains 4 green balls, 3 red balls, and 2 yellow balls. If you pick one ball at random, what is the probability of picking either a green ball or a yellow ball?

Solution: Total number of balls: 4 + 3 + 2 = 9 P(Green) = 4/9 P(Yellow) = 2/9 Since you can't pick a ball that is both green and yellow at the same time, these events are mutually exclusive.