Lesson Notes By Weeks and Term v3 - Junior Secondary 2
Probability
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Probability
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Subject: General Mathematics
Class: Junior Secondary 2
Term: 3rd Term
Week: 7
Theme: Everyday Statistics
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Discuss the occurrence of chance events in everyday life. Determine the probability of certain events Apply the occurrence of chance events / probabilities in everyday life.
number of outcomes}} = \frac{1}{2}$$ The probability of getting a Head is 1/2 or 0.
5. Example 2: Rolling a Die A fair six-sided die is rolled.
What is the probability of: a) Rolling a 4? b) Rolling an even number? c) Rolling a number greater than 6?
Step 1: Identify the Sample Space (S). S = {1, 2, 3, 4, 5, 6}. Total number of possible outcomes = 6. a)
Event: Rolling a
4. Favourable outcomes = {4}. Number of favourable outcomes = 1. $$P(\text{rolling a 4}) = \frac{1}{6}$$ b)
Event: Rolling an even number. Favourable outcomes = {2, 4, 6}. Number of favourable outcomes = 3. $$P(\text{rolling an even number}) = \frac{3}{6} = \frac{1}{2}$$ c)
Event: Rolling a number greater than
6. Favourable outcomes = { } (There are no numbers greater than 6 on a standard die). Number of favourable outcomes = 0. $$P(\text{rolling > 6}) = \frac{0}{6} = 0$$ This is an impossible event.
Example 3: Picking from a Bag (Nigerian Context) A bag contains 5 red beads, 3 blue beads, and 2 green beads, often used for local craft or counting. If a bead is picked at random, what is the probability that it is: a) A red bead? b) A green bead? c) A yellow bead?
Step 1: Identify the Sample Space (S). Total number of beads = 5 (red) + 3 (blue) + 2 (green) = 10 beads. Total number of possible outcomes = 10. a)
Event: Picking a red bead. Number of favourable outcomes (red beads) = 5. $$P(\text{red bead}) = \frac{5}{10} = \frac{1}{2}$$ b)
Event: Picking a green bead. Number of favourable outcomes (green beads) = 2. $$P(\text{green bead}) = \frac{2}{10} = \frac{1}{5}$$ c)
Event: Picking a yellow bead. Number of favourable outcomes (yellow beads) = 0. * $$P(\text{yellow bead}) = \frac{0}{10} = 0$$ This is an impossible event. This section provides a detailed explanation of the core concepts of probability, essential for the teacher's understanding and effective delivery. 2.
1. Introduction to Chance Events Definition: A chance event is an event whose outcome cannot be predicted with certainty before it happens. It involves an element of randomness.
Everyday Occurrence: Many events around us are chance events. For example, whether it will rain tomorrow, whether a commercial bus will arrive on time, whether a particular football team will win a match, or whether a specific student will be called upon by the teacher.
Examples (Nigerian Context): The likelihood of rainfall during the dry season. Winning a football betting game. Picking a specific type of 'bole' (roasted plantain) from a street vendor's tray if they have different sizes. The outcome of a 'Baba Ijebu' lottery draw. A student being selected from a class to answer a question. 2.
2. Basic Terminology in Probability To understand and calculate probabilities, several terms are fundamental: Experiment: An activity or process that has a well-defined set of possible outcomes.
Examples: Tossing a coin, rolling a die, drawing a card from a deck, picking a fruit from a basket.
Outcome: A single possible result of an experiment.
Examples: Getting a "Head" when tossing a coin, rolling a "3" on a die, drawing an "Ace" from a deck.
Sample Space (S): The set of all possible outcomes of an experiment. It is usually denoted by 'S'.
Examples: Tossing a coin: S = {Head, Tail} Rolling a standard six-sided die: S = {1, 2, 3, 4, 5, 6} Picking a student from a class of 10 boys and 12 girls: S = {all 22 students} Event (E): A subset of the sample space. It is a specific outcome or a group of outcomes that we are interested in.
Examples: "Getting a Head" when tossing a coin. "Rolling an even number" (E = {2, 4, 6}) when rolling a die. "Picking a girl" from the class.
Favourable Outcome: The outcome(s) that satisfy the condition of the event we are interested in.
Example: If the event is "rolling an even number," the favourable outcomes are {2, 4, 6}. 2.
3. The Probability Formula The probability of an event (E) occurring is calculated using the formula: $$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes (size of sample space)}}$$ Notation: P(E) stands for "the probability of Event E." Range of Probability: The probability of any event is always a number between 0 and 1, inclusive. P(E) = 0: An impossible event (e.g., rolling a 7 on a standard six-sided die). P(E) = 1: A certain event (e.g., rolling a number less than 7 on a standard six-sided die). P(E) = 0.5 (or 1/2): An event that is equally likely to happen or not happen (e.g., getting a Head when tossing a fair coin). 2.
4. Worked Examples (Illustrating Concepts and Calculations)
Example 1: Tossing a Coin An experiment involves tossing a fair coin. What is the probability of getting a Head?
Step 1: Identify the Sample Space (S). The possible outcomes are Head (H) and Tail (T). So, S = {H, T}. Total number of possible outcomes =
2. Step 2: Identify the Event (E). The event is "getting a Head." Step 3: Identify the Favourable Outcomes. The favourable outcome is {H}. Number of favourable outcomes =
1. Step 4: Apply the Probability Formula. $$P(\text{Head}) = \frac{\text{Number of Heads}}{\text{Total number of outcomes}} = \frac{1}{2}$$ The probability of getting a Head is 1/2 or 0.
5. Example 2: Rolling a Die A fair six-sided die is rolled.
What is the probability of: a) Rolling a 4? b) Rolling an even number? c) Rolling a number greater than 6?
Step 1: Identify the Sample Space (S). S = {1, 2, 3, 4, 5, 6}. Total number of possible outcomes = 6. a)
Event: Rolling a
4. Favourable outcomes = {4}. Number of favourable outcomes = 1. * $$P(\text{rolling a 4}) 3.
1. Introduction (10 minutes)
Teacher Activity: Initiate a class discussion by asking students about events whose outcomes they cannot predict with certainty.
Examples: "Will it rain later today?" "Will traffic be heavy after school?" "Will my favourite football team win this weekend?" Introduce the term "chance event" to describe these situations. Ask students to provide examples of chance events from their daily lives in Nigeria (e.g., getting a flat tire on a motorcycle, finding specific goods at the market, a NEPA power outage). Explain that mathematics provides a way to quantify these chances using "probability." Student Activity: Students brainstorm and share examples of chance events. Students listen and participate in the discussion, linking their experiences to the concept of chance. Students aim to contribute at least one relevant local example of a chance event. 3.
2. Concept Development and Explanation (25 minutes)
Teacher Activity: Terminology: Define and explain "Experiment," "Outcome," "Sample Space," "Event," and "Favourable Outcome" using simple, relatable examples.
Demonstration 1 (Coin Toss): Perform a coin toss. Ask students to list all possible outcomes (Head, Tail). Introduce "Sample Space." Ask about the outcome of interest (e.g., "getting a Head"). Introduce "Event" and "Favourable Outcome." Demonstration 2 (Die Roll): Roll a standard die. Ask for the sample space. Ask for the event "rolling an even number" and identify favourable outcomes.
Probability Formula: Introduce the formula P(E) = (Number of favourable outcomes) / (Total number of possible outcomes).
Probability Scale: Explain that probability is always between 0 and
1. Discuss "impossible events" (P=0) and "certain events" (P=1) with examples. Worked
Examples: Go through Examples 1, 2, and 3 from the "Key Concepts and Explanations" section step-by-step on the board, emphasizing how to identify the sample space, events, and favourable outcomes before applying the formula. Ensure clear, logical steps.
Student Activity: Students actively listen, take notes, and ask clarifying questions. Students participate in identifying sample spaces, events, and favourable outcomes during demonstrations. Students copy worked examples into their notebooks. Students contribute to solving parts of the examples when prompted. 3.
3. Application and Discussion of Real-Life Scenarios (10 minutes)
Teacher Activity: Lead a discussion on how probability applies to real-life situations in Nigeria, beyond the simple coin/die examples. Refer back to the "Real-World Applications" listed in the Overview. Encourage students to think critically about how understanding probability can influence daily decisions (e.g., "Should I take an umbrella today?", "What are my chances of passing if I don't study?").
Student Activity: Students brainstorm and share examples of how probability is used in Nigerian contexts (e.g., sports betting, health decisions, farming). Students discuss how understanding probability could help them make better choices. Students will work on these problems in pairs or small groups, with the teacher providing guidance and feedback. Question 1 (Targets Performance Objective 2): A bag contains 4 yellow mangoes and 6 green mangoes. If a mango is picked at random, what is the probability that it is a yellow mango?
Solution 1: Step 1: Identify the Sample Space (Total possible outcomes). Total number of mangoes = 4 (yellow) + 6 (green) = 10 mangoes.
Step 2: Identify the Event and Favourable Outcomes. Event = picking a yellow mango. Number of favourable outcomes = 4 (yellow mangoes).
Step 3: Apply the Probability Formula. $$P(\text{yellow mango}) = \frac{\text{Number of yellow mangoes}}{\text{Total number of mangoes}} = \frac{4}{10} = \frac{2}{5}$$
Commentary: This is a straightforward application of the probability formula for a simple event, similar to drawing beads from a bag. Question 2 (Targets Performance Objective 2): In a class of JSS 2, there are 25 boys and 15 girls. If a class captain is to be chosen by random ballot, what is the probability that a girl is chosen?
Solution 2: Step 1: Identify the Sample Space. Total number of students in the class = 25 (boys) + 15 (girls) = 40 students.
Step 2: Identify the Event and Favourable Outcomes. Event = choosing a girl as class captain. Number of favourable outcomes = 15 (girls).
Step 3: Apply the Probability Formula. $$P(\text{choosing a girl}) = \frac{\text{Number of girls}}{\text{Total number of students}} = \frac{15}{40} = \frac{3}{8}$$
Commentary: This example applies probability to a common school scenario, making it relatable to students. It reinforces calculating probability from given counts. Question 3 (Targets Performance Objective 1): Give three examples of chance events that commonly occur in a typical Nigerian market.
Solution 3: Finding a specific commodity: The probability of finding a particular type of fresh yam or vegetable at the market on a given day, as availability can vary.
Bargaining success: The likelihood that a buyer will successfully bargain down the price of an item with a seller.
Customer flow: The probability of a hawker making a certain number of sales within an hour.
Commentary: This question assesses the students' understanding of "chance events" and their ability to identify them in their immediate environment, meeting the first performance objective. Question 4 (Targets Performance Objective 3): A local football club, 'Enyimba FC', has played 10 matches this season. They won 6 matches, lost 2, and drew
2. Based on this historical data, what is the probability that Enyimba FC will win their next match? (Assume past performance is indicative of future probability for this exercise).
Solution 4: Step 1: Identify the Sample Space (Total trials/outcomes). Total matches played =
1
0. Step 2: Identify the Event and Favourable Outcomes. Event = Enyimba FC winning their next match. Number of favourable outcomes (wins) =
6. Step 3: Apply the Probability Formula. $$P(\text{Enyimba FC wins}) = \frac{\text{Number of wins}}{\text{Total matches played}} = \frac{6}{10} = \frac{3}{5}$$
Commentary: This question moves towards applying probability in a predictive sense, using historical data from a popular Nigerian sport, addressing the third performance objective. It also introduces the idea that probability can be estimated from observed frequencies.
This topic is highly practical and can be integrated into various aspects of Nigerian life: Agriculture and Environmental Science: Application: Farmers in Nigeria rely heavily on rainfall for crop cultivation. Probability is used to predict the likelihood of rain, drought, or floods based on historical weather patterns. This helps them decide on planting times, crop types (e.g., drought-resistant varieties), and irrigation needs. For example, farmers in the North might consider the probability of early cessation of rains before planting crops with long maturity periods.
Integration: Discuss with students how local farmers make decisions influenced by weather. Bring in local news reports or traditional proverbs related to weather prediction.
Business and Entrepreneurship: Application: Small business owners (e.g., 'mama put' owners, 'okada' riders, shopkeepers) implicitly use probability. A restaurateur might estimate the probability of selling out a particular dish based on customer preferences and past sales data to avoid waste or stockouts. An 'okada' rider might consider the probability of finding passengers in different routes or times of day.
Integration: Ask students to interview a local business owner (if feasible, or discuss hypothetically) about how they make daily decisions that involve an element of uncertainty.
Connect it to probability: "What's the chance you'll sell all your goods today?" Health and Safety: Application: Public health officials use probability to assess the risk of disease outbreaks (e.g., cholera, malaria) in communities, especially during certain seasons. This helps in allocating resources for vaccination campaigns or sanitation improvements. Individuals also use it when assessing the probability of road accidents or the effectiveness of certain safety measures (e.g., wearing a helmet).
Integration: Discuss with students how health campaigns (e.g., against malaria) consider the probability of infection. Talk about road safety awareness and how obeying traffic rules reduces the probability of accidents.