Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Apply the addition rule for different types of events.
• Solve problems with independent events using diagrams and tables.
• Calculate conditional probabilities.
• Use Venn diagrams to represent and calculate probabilities.

Lesson summary

This week, we delve deeper into the fascinating world of probability, building upon the foundational knowledge you gained in previous grades. Probability is not just an abstract mathematical concept; it plays a crucial role in understanding and navigating the uncertainties of daily life. From predicting the likelihood of rain for a Limpopo farmer planning their crops, to understanding the odds in the Lotto, probability is a powerful tool for informed decision-making.

Lesson notes

2. 1.

The Addition Rule: The addition rule helps us find the probability of event A or event B occurring.

There are two main scenarios: Mutually Exclusive Events: These events cannot happen at the same time. For example, you can't roll a 3 and a 6 on a single roll of a die.

The formula is: P(A or B) = P(A) + P(B)

Non-Mutually Exclusive Events: These events can happen at the same time. For example, drawing a heart or a king from a deck of cards. You can draw the King of Hearts!

The formula is: P(A or B) = P(A) + P(B) – P(A and B) The P(A and B) term is crucial to avoid double-counting the outcomes that belong to both A and

B. Example 1 (Mutually Exclusive): A bag contains 5 red balls and 3 blue balls. What is the probability of drawing a red ball or a blue ball? P(Red) = 5/8 P(Blue) = 3/8 P(Red or Blue) = P(Red) + P(Blue) = 5/8 + 3/8 = 1 This makes sense because you are guaranteed to draw either a red or a blue ball.

Example 2 (Non-Mutually Exclusive): In a class of 30 learners, 15 play soccer, 10 play rugby, and 5 play both. What is the probability that a learner selected at random plays soccer or rugby? P(Soccer) = 15/30 P(Rugby) = 10/30 P(Soccer and Rugby) = 5/30 P(Soccer or Rugby) = P(Soccer) + P(Rugby) – P(Soccer and Rugby) = 15/30 + 10/30 – 5/30 = 20/30 = 2/3 2.

2. Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin and rolling a die are independent events. The outcome of the coin flip doesn't change the probability of any particular number appearing on the die.

Formula: P(A and B) = P(A) P(B) This formula is KEY to determining if events are independent. If the equation holds true, the events are independent. If it doesn't, they are dependent.

Example 3 (Independent): 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. Conditional Probability: Conditional probability is the probability of an event A occurring, given that another event B has already occurred. We write this as P(A|B), which reads "the probability of A given B." Formula: P(A|B) = P(A and B) / P(B) This formula can be rearranged to calculate P(A and B) if P(A|B) and P(B) are known.

Example 4 (Conditional Probability): A survey was conducted among 100 students at a Johannesburg university. 60 students like Mango juice, and 40 students like both Mango and Orange juice. Given that a student likes Mango juice, what is the probability that they also like Orange juice? Let M = event that a student likes Mango juice Let O = event that a student likes Orange juice P(M) = 60/100 = 0.6 P(M and O) = 40/100 = 0.4 We want to find P(O|M) = P(O and M) / P(M) = 0.4 / 0.6 = 2/3 2.

4. Tree Diagrams and Contingency Tables: These are visual tools that help organize and calculate probabilities, especially when dealing with sequential events or conditional probabilities.

Tree Diagrams: Branching diagrams that show the possible outcomes of a series of events. Each branch represents a possible outcome, and the probabilities are written along the branches.

Contingency Tables: Tables that display the frequencies of different categories of data. They are particularly useful for calculating conditional probabilities.

Example 5 (Tree Diagram): A factory produces light bulbs. 80% of the bulbs are good (G) and 20% are defective (D). A quality control inspector tests each bulb. The probability that a good bulb passes the test is 95%, and the probability that a defective bulb fails the test is 90%. What is the probability that a bulb is defective and passes the test?

First Branch: 80% Good (0.8), 20% Defective (0.2)

Second Branch (from Good): 95% Pass (0.95), 5% Fail (0.05)

Second Branch (from Defective): 10% Pass (0.1), 90% Fail (0.9) To find P(Defective and Pass), we follow the branch "Defective" then "Pass": 0.2 * 0.1 = 0.02 or 2% 2.

5. Venn Diagrams: Venn diagrams are graphical representations used to show relationships between sets. In probability, they help visualize events and their probabilities. The overlapping regions represent the intersection of events (A and B), while the entire diagram represents the sample space.

Example 6 (Venn Diagram): Let's use the Soccer/Rugby example from earlier. Draw a Venn Diagram. Let the circle on the left be soccer and the circle on the right be rugby. The overlapping region is 5 (those who play both).

Soccer only: 15-5 =

1

0. Rugby only: 10-5 =

5. Outside both circles is 30- (10+5+5) =

1

0. The Venn diagram clearly represents the number of learners in each category, enabling easy probability calculations. Guided Practice (With Solutions)

Question 1: A standard deck of 52 cards is shuffled. What is the probability of drawing a heart or a face card (Jack, Queen, or King)?