Lesson Notes By Weeks and Term v5 - Grade 10
Probability – Week 1 focus
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Probability – Week 1 focus
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Subject: Mathematics
Class: Grade 10
Term: Term 4
Week: 1
Theme: General lesson support
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Probability is the branch of mathematics that deals with the likelihood of an event occurring. In simpler terms, it's about understanding how likely something is to happen. This concept is fundamental not only in mathematics but also in various real-life scenarios. Understanding probability allows us to make informed decisions, assess risks, and interpret data more effectively. For South African learners, probability concepts are relevant in areas such as understanding weather forecasts (likelihood of rain impacting farming), analyzing sports statistics (predicting team wins), evaluating the lottery (calculating chances of winning), and even understanding the spread of diseases (predicting...
2.1 Basic Definitions Experiment: An activity or process whose outcome is uncertain. Examples include tossing a coin, rolling a die, or drawing a card from a deck.
Sample Space (S): The set of all possible outcomes of an experiment. For example, if we toss a coin, the sample space is S = {Heads, Tails}. If we roll a standard six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}.
Event (E): A subset of the sample space. An event is a specific outcome or a set of outcomes that we are interested in. For example, if we roll a die, the event "rolling an even number" is E = {2, 4, 6}.
Outcome: A single possible result of an experiment. For example, if we toss a coin, "Heads" is an outcome.
Probability (P(E)): A measure of how likely an event E is to occur. It is a number between 0 and 1 (inclusive), where 0 indicates impossibility and 1 indicates certainty. 2.2 Calculating Probability The basic formula for calculating the probability of an event E is: P(E) = (Number of favorable outcomes for event E) / (Total number of possible outcomes in the sample space S)
Important Notes: The probability of any event must be between 0 and
1. The sum of the probabilities of all possible outcomes in the sample space must equal 1. 2.3 Representing Sample Spaces Listing: Simply listing all possible outcomes within curly braces {}.
Example: Rolling a die: S = {1, 2, 3, 4, 5, 6}.
Tree Diagrams: Useful for experiments with multiple stages. Each branch represents a possible outcome.
Two-Way Tables: Used to organize outcomes when two or more variables are involved. 2.4 Mutually Exclusive and Non-Mutually Exclusive Events Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. In other words, they have no outcomes in common. For example, when tossing a coin, you can either get heads or tails, but not both at the same time.
Therefore, the events "getting heads" and "getting tails" are mutually exclusive. If events A and B are mutually exclusive, P(A and B) =
0. Non-Mutually Exclusive Events: Two events are non-mutually exclusive if they can occur at the same time. They have outcomes in common. For example, when rolling a die, the event "rolling an even number" (E = {2, 4, 6}) and the event "rolling a number greater than 3" (G = {4, 5, 6}) are non-mutually exclusive because they both include the outcomes 4 and 6. 2.5 Worked Examples Example 1: Tossing a Coin A fair coin is tossed once. What is the probability of getting heads?
Sample space: S = {Heads, Tails} Event: E = {Heads} Number of favorable outcomes for E: 1 Total number of possible outcomes: 2 P(Heads) = 1/2 = 0.5 or 50% Example 2: Rolling a Die A fair six-sided die is rolled once. What is the probability of rolling a number greater than 4?
Sample space: S = {1, 2, 3, 4, 5, 6} Event: E = {5, 6} Number of favorable outcomes for E: 2 Total number of possible outcomes: 6 P(Rolling a number greater than 4) = 2/6 = 1/3 or approximately 33.33% Example 3: Drawing a Card A card is drawn randomly from a standard deck of 52 playing cards. What is the probability of drawing a heart?
Sample space: S = {52 cards} Event: E = {drawing a heart} Number of favorable outcomes for E: 13 (there are 13 hearts in a deck)
Total number of possible outcomes: 52 P(drawing a heart) = 13/52 = 1/4 = 0.25 or 25% Example 4: Learners in a Class In a class of 30 learners, 12 play soccer, 8 play netball, and 5 play both. What is the probability that a learner chosen at random plays soccer?
Total learners: 30 Learners who play Soccer: 12 P(Learner plays Soccer) = 12/30 = 2/5 = 0.4 or 40% Example 5: Using a Tree Diagram A bag contains 3 red balls and 2 blue balls. A ball is drawn, its colour noted, and then replaced. A second ball is then drawn. Draw a tree diagram to represent the possible outcomes and calculate the probability of drawing two red balls.
Step 1: Draw the tree diagram.
First Draw: Branch out to Red (R) and Blue (B). P(R) = 3/5, P(B) = 2/5 Second Draw (from each branch of the first draw, as the ball is replaced): Branch out to Red (R) and Blue (B). Again, P(R) = 3/5, P(B) = 2/
5. Step 2: Identify the outcome of interest: We want two red balls (R,R).
Step 3: Calculate the probability: P(R,R) = P(R on first draw) P(R on second draw) = (3/5) * (3/5) = 9/
2
5. Guided Practice (With Solutions)
Question 1: A bag contains 4 green marbles and 6 yellow marbles. What is the probability of randomly selecting a green marble?
Solution: Number of green marbles: 4 Total number of marbles: 4 + 6 = 10 P(Green) = 4/10 = 2/5 = 0.4 or 40%.
Commentary: This is a straightforward application of the basic probability formula. We identify the number of favorable outcomes (green marbles) and divide it by the total number of possible outcomes (all marbles).
Question 2: A spinner is divided into 8 equal sections, numbered 1 to
8. What is the probability of spinning an odd number?