Lesson Notes By Weeks and Term v5 - Grade 10
Probability – Week 4 focus
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Probability – Week 4 focus
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Subject: Mathematics
Class: Grade 10
Term: Term 4
Week: 4
Theme: General lesson support
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Probability is the study of chance and uncertainty. It's a vital tool for understanding and predicting the likelihood of events, from the simple toss of a coin to more complex scenarios like weather forecasting or the spread of diseases. In South Africa, probability is used in various sectors, including insurance (calculating risk), gambling (predicting outcomes), and even in agricultural planning (estimating crop yields based on weather patterns). Understanding probability empowers learners to make informed decisions in their everyday lives, critically analyse information, and develop problem-solving skills applicable to diverse situations.
2.1 Basic Probability: Probability is the measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Experiment: Any process with well-defined outcomes. (e.g., tossing a coin, rolling a die).
Outcome: A possible result of an experiment. (e.g., heads when tossing a coin, 3 when rolling a die).
Sample Space (S): The set of all possible outcomes of an experiment. (e.g., {Heads, Tails} for a coin toss, {1, 2, 3, 4, 5, 6} for a die roll).
Event (E): A subset of the sample space. (e.g., getting an even number when rolling a die, E = {2, 4, 6}).
Probability of an Event: The probability of an event E occurring is given by: P(E) = (Number of favourable outcomes for E) / (Total number of possible outcomes in S) = n(E)/n(S)
Example 1: A bag contains 5 red balls and 3 blue balls. What is the probability of randomly selecting a red ball?
Event (E): Selecting a red ball. Number of favourable outcomes n(E) = 5 Total number of possible outcomes n(S) = 5 + 3 = 8 P(E) = 5/8 Therefore, the probability of selecting a red ball is 5/8. 2.2 Compound Events: Compound events involve two or more events happening together or in sequence.
Independent Events: Two events are independent if the outcome of one event does not affect the outcome of the other. P(A and B) = P(A)
P(B)
Dependent Events: Two events are dependent if the outcome of one event affects the outcome of the other. P(A and B) = P(A) P(B|A), where P(B|A) is the probability of B occurring given that A has already occurred.
Example 2: Independent Events A coin is tossed 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 Therefore, the probability of getting heads and a 4 is 1/
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2. Example 3: Dependent Events A bag contains 5 red balls and 3 blue balls. Two balls are drawn without replacement. What is the probability of drawing a red ball first and then another red ball? P(Red first) = 5/8 P(Red second | Red first) = 4/7 (since one red ball has been removed) P(Red first and Red second) = (5/8) (4/7) = 20/56 = 5/14 Therefore, the probability of drawing two red balls in a row is 5/14. 2.3 Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. For example, when tossing a coin, you cannot get both heads and tails simultaneously. Addition Rule for Mutually Exclusive Events: P(A or B) = P(A) + P(B)
Example 4: A die is rolled. What is the probability of rolling a 2 or a 5? P(2) = 1/6 P(5) = 1/6 P(2 or 5) = P(2) + P(5) = (1/6) + (1/6) = 2/6 = 1/3 Therefore, the probability of rolling a 2 or a 5 is 1/3. 2.4 Venn Diagrams and Probability: Venn diagrams are useful for visualizing sets and their relationships. They can be used to calculate probabilities involving combined events, especially when events are not mutually exclusive. P(A or B) = P(A) + P(B) - P(A and B)
Example 5: In a class of 30 learners, 15 play soccer, 10 play rugby, and 5 play both. What is the probability that a randomly selected learner plays soccer or rugby? P(Soccer) = 15/30 = 1/2 P(Rugby) = 10/30 = 1/3 P(Soccer and Rugby) = 5/30 = 1/6 P(Soccer or Rugby) = P(Soccer) + P(Rugby) - P(Soccer and Rugby) = (1/2) + (1/3) - (1/6) = 3/6 + 2/6 - 1/6 = 4/6 = 2/3 Therefore, the probability that a randomly selected learner plays soccer or rugby is 2/3. 2.5 Tree Diagrams and Two-Way Tables These are visual aids for outlining the possible outcomes of a compound event, especially when the number of outcomes is limited. They enable one to systematically calculate the probability of various combinations of outcomes.
Example 6: Tree Diagram A coin is tossed twice. Draw a tree diagram to show the possible outcomes. ``` Coin Toss 1 Coin Toss 2 Outcome / \ / \ H T H T HH, HT, TH, TT / \ / \ H T H T ``` Using the tree diagram, we can calculate P(Two Heads) = 1/4, P(One Head and One Tail) = 2/4 = 1/2 Example 7: Two-Way Table A survey of 100 students revealed the following information about their participation in sports and clubs: | | Sports | No Sports | Total | |-------------|--------|-----------|-------| | Clubs | 25 | 30 | 55 | | No Clubs | 20 | 25 | 45 | | Total | 45 | 55 | 100 | What is the probability a student participates in both sports and clubs? P(Sports and Clubs) = 25/100 = 1/4 Guided Practice (With Solutions)
Question 1: A standard six-sided die is rolled. What is the probability of rolling an odd number?
Solution: Sample Space (S) = {1, 2, 3, 4, 5, 6} Event (E): Rolling an odd number = {1, 3, 5} n(E) = 3 n(S) = 6 P(E) = n(E)/n(S) = 3/6 = 1/2 Therefore, the probability of rolling an odd number is 1/
2. Question 2: A box contains 4 green apples and 6 red apples. Two apples are drawn at random, without replacement. What is the probability that both apples are red?