Lesson Notes By Weeks and Term v5 - Grade 11
Probability – Week 10 focus
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Probability – Week 10 focus
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Subject: Mathematics
Class: Grade 11
Term: 3rd Term
Week: 10
Theme: General lesson support
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Probability is the study of the likelihood of events occurring. It's a fundamental concept in mathematics that has wide-ranging applications in everyday life. From predicting the outcome of a soccer match between Kaizer Chiefs and Orlando Pirates to assessing the risks associated with investments, probability helps us make informed decisions in the face of uncertainty. In South Africa, understanding probability is crucial for interpreting statistical data related to health, economics, and social issues, empowering citizens to engage critically with information presented to them.
2.1 Compound Events: A compound event is an event that consists of two or more simple events happening together. For example, tossing a coin twice and getting heads on both tosses is a compound event. We'll look at two main methods for calculating probabilities of compound events: Tree Diagrams and Contingency Tables. 2.2 Tree Diagrams: Tree diagrams are a visual way to represent the possible outcomes of a sequence of events. Each branch of the tree represents a possible outcome, and the probability of that outcome is written along the branch. To find the probability of a particular sequence of events, you multiply the probabilities along the corresponding branches.
Example 1: A bag contains 3 red marbles and 2 blue marbles. A marble is drawn at random, its color is noted, and then it is replaced. A second marble is then drawn. What is the probability of drawing a red marble followed by a blue marble?
Step 1: Draw the tree diagram.
First Draw: Branch 1: Red (R), Probability = 3/5 Branch 2: Blue (B), Probability = 2/5 Second Draw (emanating from each branch of the first draw): From Red (R): Branch 1: Red (R), Probability = 3/5 Branch 2: Blue (B), Probability = 2/5 From Blue (B): Branch 1: Red (R), Probability = 3/5 Branch 2: Blue (B), Probability = 2/5 Step 2: Identify the path for the desired outcome (Red then Blue). The path we want is Red (from the first draw) followed by Blue (from the second draw).
Step 3: Multiply the probabilities along that path. P(Red then Blue) = P(Red on 1st draw) P(Blue on 2nd draw) = (3/5) * (2/5) = 6/25 Therefore, the probability of drawing a red marble followed by a blue marble is 6/
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5. Example 2: A biased coin has a probability of 0.6 of landing heads. The coin is tossed twice. What is the probability of getting at least one tail?
Step 1: Draw the tree diagram.
First Toss: Branch 1: Heads (H), Probability = 0.6 Branch 2: Tails (T), Probability = 0.4 Second Toss (emanating from each branch of the first toss): From Heads (H): Branch 1: Heads (H), Probability = 0.6 Branch 2: Tails (T), Probability = 0.4 From Tails (T): Branch 1: Heads (H), Probability = 0.6 Branch 2: Tails (T), Probability = 0.4 Step 2: Identify the paths for "at least one tail". Heads then Tails (HT) Tails then Heads (TH)
Tails then Tails (TT)
Step 3: Calculate the probability of each path and add them together. P(HT) = 0.6 0.4 = 0.24 P(TH) = 0.4 0.6 = 0.24 P(TT) = 0.4 0.4 = 0.16 P(at least one tail) = 0.24 + 0.24 + 0.16 = 0.64 Alternatively, we could find the probability of no tails (HH) which is 0.6 * 0.6 = 0.
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6. Then, P(at least one tail) = 1 - P(no tails) = 1 - 0.36 = 0.64. 2.3 Contingency Tables: A contingency table (also called a two-way table) is a table that displays the frequency distribution of two or more categorical variables. These tables are very useful for calculating conditional probabilities.
Example 3: A survey was conducted at a high school to determine the relationship between smoking and playing sports. The results are shown in the contingency table below: | | Smoker | Non-Smoker | Total | |----------------|--------|------------|-------| | Plays Sports | 10 | 90 | 100 | | Does Not Play Sports | 40 | 60 | 100 | | Total | 50 | 150 | 200 | What is the probability that a student chosen at random: a) Smokes? b) Plays Sports? c) Smokes and Plays Sports? d) Smokes, given that they Play Sports? e) Plays Sports, given that they Smoke?
Solutions: a) P(Smokes) = Total Smokers / Total Students = 50/200 = 1/4 = 0.25 b) P(Plays Sports) = Total Who Play Sports / Total Students = 100/200 = 1/2 = 0.5 c) P(Smokes and Plays Sports) = Number Who Smoke AND Play Sports / Total Students = 10/200 = 1/20 = 0.05 d) P(Smokes | Plays Sports) = Number Who Smoke AND Play Sports / Total Who Play Sports = 10/100 = 1/10 = 0.1 (This is read as "The probability of smoking given that they play sports.") e) P(Plays Sports | Smokes) = Number Who Play Sports AND Smoke / Total Smokers = 10/50 = 1/5 = 0.2 (This is read as "The probability of playing sports given that they smoke.") 2.4 Mutually Exclusive Events: Two events are mutually exclusive if they cannot both occur at the same time. For example, tossing a coin once: getting heads and getting tails are mutually exclusive. Addition Rule for Mutually Exclusive Events: If events A and B are mutually exclusive, then P(A or B) = P(A) + P(B).
Example 4: A box contains 5 green balls, 3 blue balls and 2 red balls. A ball is chosen at random from the box. Find the probability that the ball is either green or red. P(Green) = 5/10 = 1/2 P(Red) = 2/10 = 1/5 Since the ball cannot be both green and red at the same time (mutually exclusive), P(Green or Red) = P(Green) + P(Red) = 1/2 + 1/5 = 5/10 + 2/10 = 7/10. 2.5 Independent Events: Two events are independent if the outcome of one event does not affect the outcome of the other. For example, tossing a coin and rolling a die. Multiplication Rule for Independent Events: If events A and B are independent, then P(A and B) = P(A) * P(B).