Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate probabilities of single and combined events.
• Interpret probability values in terms of risk.
• Use probability to make predictions.
• Differentiate between independent and dependent events.
• Calculate expected values for financial decisions.

Lesson summary

This week, we delve into the world of probability, focusing on how we can use it to predict outcomes and assess risk. Understanding probability is crucial for making informed decisions in various aspects of life, from financial planning and health choices to understanding weather forecasts and the odds in sports betting. In South Africa, where socio-economic disparities can significantly impact opportunities and vulnerabilities, a solid understanding of probability allows learners to better navigate uncertainties, evaluate risks, and make sound judgments.

Lesson notes

2.1 What is Probability? Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, inclusive. 0: The event is impossible (it will never happen). 1: The event is certain (it will always happen).

Values between 0 and 1: Represent varying degrees of likelihood. For instance, 0.5 (or 50%) indicates an equal chance of the event occurring or not occurring. Probability can be expressed as a fraction, a decimal, or a percentage. 2.2 Basic Probability Formula: The probability of an event (P(Event)) is calculated as: P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes)

Example: What is the probability of rolling a '4' on a standard six-sided die?

Number of favourable outcomes: 1 (only one side has a '4')

Total number of possible outcomes: 6 (six sides on the die) P(rolling a '4') = 1/6 ≈ 0.167 or 16.7% 2.3 Sample Space: The sample space is the set of all possible outcomes of an experiment.

Example: The sample space for flipping a coin is {Heads, Tails}. The sample space for rolling a six-sided die is {1, 2, 3, 4, 5, 6}. 2.4 Types of Events: Independent Events: Events where the outcome of one event does not affect the outcome of another event. For example, flipping a coin and rolling a die are independent events. The result of the coin flip doesn't change the possible outcomes of the die roll.

Dependent Events: Events where the outcome of one event does affect the outcome of another event. For example, drawing two cards from a deck without replacement. The first card you draw changes the composition of the remaining deck, affecting the probability of what you draw next.

Mutually Exclusive Events: Events that cannot occur at the same time. For example, flipping a coin – you can't get heads and tails simultaneously. 2.5 Calculating Probabilities of Multiple Events: 'AND' Rule (for Independent Events): The probability of two independent events A and B both occurring is: P(A and B) = P(A) * P(B)

Example: What is the probability of flipping a coin and getting heads, AND rolling a die and getting a 6? P(Heads) = 1/2 P(6) = 1/6 P(Heads and 6) = (1/2) * (1/6) = 1/12 ≈ 0.083 or 8.3% 'OR' Rule (for Mutually Exclusive Events): The probability of event A OR event B occurring (when they are mutually exclusive) is: P(A or B) = P(A) + P(B)

Example: What is the probability of rolling a 1 OR a 2 on a six-sided die? P(1) = 1/6 P(2) = 1/6 P(1 or 2) = (1/6) + (1/6) = 2/6 = 1/3 ≈ 0.333 or 33.3% 'OR' Rule (for Non-Mutually Exclusive Events): The probability of event A OR event B occurring (when they are not mutually exclusive) is: P(A or B) = P(A) + P(B) - P(A and B)

Example: Consider a class of 30 learners. 15 study Maths, 10 study Science, and 5 study both. What is the probability that a randomly selected learner studies Maths OR Science? P(Maths) = 15/30 P(Science) = 10/30 P(Maths and Science) = 5/30 P(Maths or Science) = (15/30) + (10/30) - (5/30) = 20/30 = 2/3 ≈ 0.667 or 66.7% 2.6 Tree Diagrams: Tree diagrams are useful for visualizing and calculating probabilities in multi-step experiments. Each branch represents a possible outcome, and the probabilities along each branch are multiplied to find the probability of that sequence of events.

Example: A bag contains 3 red balls and 2 blue balls. A ball is drawn at random and not replaced. Then, a second ball is drawn. What is the probability of drawing two red balls?

First Draw: P(Red) = 3/5 P(Blue) = 2/5 Second Draw (dependent on the first draw): If the first ball was red: P(Red) = 2/4 P(Blue) = 2/4 If the first ball was blue: P(Red) = 3/4 P(Blue) = 1/4 The probability of drawing two red balls is (3/5) * (2/4) = 6/20 = 3/10 = 0.3 or 30%. 2.7 Venn Diagrams: Venn diagrams are visual tools used to represent sets and their relationships. They can be helpful in calculating probabilities involving unions and intersections of events. See the earlier example of Maths and Science students using the 'OR' rule for Non-Mutually Exclusive Events, the calculations could also be visualised using Venn diagrams. 2.8 Expected Value Expected value is the average outcome we expect if we repeat an experiment a large number of times. It is calculated by multiplying each possible outcome by its probability and summing the results.

Formula: Expected Value = (Outcome 1 Probability of Outcome 1) + (Outcome 2 Probability of Outcome 2) + ...

Example: Consider a lottery ticket that costs R

5. The probability of winning R100 is 1/100, and the probability of winning nothing is 99/

1

0

0. What is the expected value of buying a ticket?

Outcome 1: Winning R100 (net win of R95 after deducting the ticket cost)

Probability of Outcome 1: 1/100 Outcome 2: Winning nothing (net loss of R5)

Probability of Outcome 2: 99/100 Expected Value = (R95 1/100) + (-R5 99/100) = R0.95 - R4.95 = -R4.00 This means that, on average, you would expect to lose R4.00 for every lottery ticket you buy. This highlights the risk associated with gambling.