Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Probability
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Probability
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Subject: Further Mathematics
Class: Senior Secondary 2
Term: 1st Term
Week: 2
Theme: Statistics
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Under stand the frequential, classiscal and axiomatic approach to probability Define some terms used in probability (e.g random experiment, sample space,events e.t.c) Solve problems on dependent and in dependent events Solve problems on complements of events. Solve problems on conditional probability. Under stand and use probability trees in solving problems.
This section provides in-depth explanations and examples for all core concepts. This section outlines practical activities for effective lesson delivery.
A. Teacher Activities: Introduction (10 minutes): Begin by asking students to brainstorm situations in their daily lives or in Nigeria where uncertainty exists (e.g., predicting exam questions, football match results, traffic jams, winning a lottery). Introduce the concept of probability as a way to quantify this uncertainty. State the learning objectives for the lesson. Concept Explanation & Discussion (25 minutes): Approaches: Explain the Frequential, Classical, and Axiomatic approaches using local examples (e.g., historical data on electricity blackouts for frequential, fair dice/coins for classical, basic rules for axiomatic). Encourage students to provide their own examples.
Definitions: Clearly define key terms: random experiment, outcome, sample space, event. Use simple experiments (tossing a coin, rolling a die, drawing names from a hat) to illustrate these terms. Guide students to identify sample spaces and events. Dependent vs.
Independent Events (20 minutes): Explain independent events with examples like tossing a coin multiple times or rolling two dice. Demonstrate the multiplication rule. Explain dependent events using "without replacement" scenarios (e.g., drawing two different coloured markers from a box, selecting two students from a class for different roles). Demonstrate the multiplication rule for dependent events. Facilitate a short activity where students identify whether given scenarios represent independent or dependent events. Complements and Conditional Probability (20 minutes): Explain the concept of the complement of an event and its formula P(E') = 1 - P(E) using practical examples (e.g., probability of rain vs. no rain, passing vs. failing). Introduce conditional probability using the formula P(A|B) = P(A and B) / P(B). Use a small class data set (e.g., number of students who play football and basketball) to illustrate.
Probability Trees (25 minutes): Demonstrate the construction of a probability tree diagram step-by-step using a practical Nigerian scenario (e.g., two stages of selection for a scholarship, weather outcomes affecting crop yield). Show how to calculate probabilities of individual paths and combined events using the tree. Guided Practice & Problem Solving (Remaining time/Integrated): Work through guided practice problems (as provided in Section 4) on the board, encouraging student input at each step. Correct misconceptions immediately.
Wrap-up & Assignment: Summarize key concepts covered. Assign independent practice questions (from Section 5) for homework.
B. Student Activities: Brainstorming: Students brainstorm real-life situations involving uncertainty.
Definition & Identification: Students work in pairs to define the key terms in their own words after teacher explanation. Given simple experiments, students identify outcomes, sample spaces, and various events.
Scenario Analysis: Students classify given scenarios as involving independent or dependent events, justifying their answers.
Problem Solving (Group/Individual): Students solve problems on independent, dependent, mutually exclusive, conditional probability, and complements, first in groups, then individually. Students participate actively during guided practice, suggesting steps and solutions.
Tree Diagram Construction: Students attempt to construct probability trees for given two-stage or three-stage experiments.
Presentation: Selected students present their solutions to the class, explaining their reasoning.
Materials: Whiteboard/Chalkboard, markers/chalk, projector (optional), dice, coins, flashcards with probability terms, simple real-life items for demonstration (e.g., different coloured pens in a bag). This section provides worked examples to reinforce understanding. Question 1 (Basic Probability, Mutually Exclusive, Complement): In a basket at a market stall, there are 15 ripe mangoes, 10 unripe mangoes, and 5 spoilt mangoes. A customer randomly selects one mango. a) What is the probability that the mango is ripe? b) What is the probability that the mango is either unripe or spoilt? c) What is the probability that the mango is not spoilt?
Solution 1: Total number of mangoes = 15 (ripe) + 10 (unripe) + 5 (spoilt) = 30 mangoes. a) Let R be the event that the mango is ripe. Number of ripe mangoes =
1
5. P(R) = (Number of ripe mangoes) / (Total number of mangoes) = 15/30 = 1/2 or 0.5. b) Let U be the event that the mango is unripe, and S be the event that the mango is spoilt. Number of unripe mangoes =
1
0. Number of spoilt mangoes =
5. Events U and S are mutually exclusive (a mango cannot be both unripe and spoilt at the same time). P(U or S) = P(U) + P(S) = (10/30) + (5/30) = 15/30 = 1/2 or 0.5. c) Let S be the event that the mango is spoilt. The event "not spoilt" is the complement of S, denoted S'. P(S) = 5/30 = 1/
6. P(S') = 1 - P(S) = 1 - (1/6) = 5/
6. Alternatively, (Number of not spoilt mangoes) / (Total number of mangoes) = (15 + 10) / 30 = 25/30 = 5/
6. Question 2 (Independent Events): A student is preparing for two tests: Further Maths and Economics. The probability of passing Further Maths is 0.7, and the probability of passing Economics is 0.
8. Assume the outcomes of the tests are independent. a) What is the probability that the student passes both tests? b) What is the probability that the student passes Further Maths but fails Economics?
Solution 2: Let FM be the event of passing Further Maths, and ECO be the event of passing Economics. P(FM) = 0.7, P(ECO) = 0.
8. Since the events are independent: a) Probability of passing both tests = P(FM and ECO) = P(FM) × P(ECO) P(FM and ECO) = 0.7 × 0.8 = 0.56. b) Probability of failing Economics = P(ECO') = 1 - P(ECO) = 1 - 0.8 = 0.
2. Probability of passing Further Maths and failing Economics = P(FM and ECO') = P(FM) × P(ECO') (due to independence) P(FM and ECO') = 0.7 × 0.2 = 0.
1
4. Question 3 (Dependent Events & Conditional Probability): In a class of 40 students, 18 are boys and 22 are girls. The teacher wants to select two students randomly without replacement to represent the class at a debate competition. a) What is the probability that the first student selected is a boy and the second student selected is also a boy? b) What is the probability that the first student selected is a boy, and the second student selected is a girl?
Solution 3: Total students =
4
0. Number of boys = 18, Number of girls =
2
2. Events are dependent because selection is without replacement. a) Let B1 be the event the first student is a boy, and B2 be the event the second student is a boy. P(B1) = 18/
4
0. After selecting one boy, there are now 17 boys left and a total of 39 students. P(B2|B1) = 17/39 (probability of selecting a second boy given the first was a boy). P(B1 and B2) = P(B1) × P(B2|B1) = (18/40) × (17/39) = (9/20) × (17/39) = (3/20) × (17/13) = 51/
2
6
0. P(B1 and B2) = 51/260 (approx. 0.196). b) Let G2 be the event the second student is a girl. P(B1) = 18/
4
0. After selecting one boy, there are still 22 girls left, and a total of 39 students. P(G2|B1) = 22/39 (probability of selecting a girl given the first was a boy). P(B1 and G2) = P(B1) × P(G2|B1) = (18/40) × (22/39) = (9/20) × (22/39) = (9/10) × (11/39) = 99/390 = 33/130 (approx. 0.254).
Question 4 (Probability Tree): A small-scale farmer in Abia State determines that there is a 40% and B2) = 51/260 (approx. 0.196). b) Let G2 be the event the second student is a girl. P(B1) = 18/
4
0. After selecting one boy, there are still 22 girls left, and a total of 39 students. P(G2|B1) = 22/39 (probability of selecting a girl given the first was a boy). P(B1 and G2) = P(B1) × P(G2|B1) = (18/40) × (22/39) = (9/20) × (22/39) = (9/10) × (11/39) = 99/390 = 33/130 (approx. 0.254).
Question 4 (Probability Tree): A small-scale farmer in Abia State determines that there is a 40% chance of a pest infestation in the next planting season. If there is a pest infestation, the probability of a significant crop loss is 0.
7. If there is no pest infestation, the probability of a significant crop loss is 0.1. a) Draw a probability tree diagram to represent this information. b) Calculate the probability of a significant crop loss. c) Calculate the probability of no significant crop loss.
Solution 4: Let I be the event of pest infestation, I' be no pest infestation. Let L be the event of significant crop loss, L' be no significant crop loss.
Given: P(I) = 0.4, P(I') = 1 - 0.4 = 0.
6. P(L|I) = 0.7, P(L'|I) = 1 - 0.7 = 0.
3. P(L|I') = 0.1, P(L'|I') = 1 - 0.1 = 0.9. a)
Probability Tree Diagram: ``` Start | |--- P(I) = 0.4 --- (Infestation) --- P(L|I) = 0.7 --- (Loss) => P(I and L) = 0.4 0.7 = 0.28 | | | |--- P(L'|I) = 0.3 --- (No Loss) => P(I and L') = 0.4 0.3 = 0.12 | |--- P(I') = 0.6 --- (No Infestation) --- P(L|I') = 0.1 --- (Loss) => P(I' and L) = 0.6 0.1 = 0.06 | |--- P(L'|I') = 0.9 --- (No Loss) => P(I' and L') = 0.6 0.9 = 0.54 ``` b) Probability of a significant crop loss (P(L)) is the sum of probabilities of paths leading to Loss. P(L) = P(I and L) + P(I' and L) P(L) = (0.4 × 0.7) + (0.6 × 0.1) = 0.28 + 0.06 = 0.34. c) Probability of no significant crop loss (P(L')) is the sum of probabilities of paths leading to No Loss. P(L') = P(I and L') + P(I' and L') P(L') = (0.4 × 0.3) + (0.6 × 0.9) = 0.12 + 0.54 = 0.66. (Alternatively, P(L') = 1 - P(L) = 1 - 0.34 = 0.66).
Public Health Decisions (Nigerian Context): Application: Public health officials use probability to assess the risk of disease outbreaks (e.g., cholera, Lassa fever, meningitis) in different regions of Nigeria. They also use it to determine the effectiveness of vaccination campaigns or public awareness programs. For instance, the probability of contracting malaria can be assessed based on environmental factors (e.g., proximity to stagnant water) and personal preventative measures.
Integration: Students can research local health statistics, for example, the number of confirmed cases of a specific disease in their state over a period, and use the frequential approach to estimate the probability of future cases. They can also discuss how conditional probability might be used to determine the likelihood of contracting a disease given certain risk factors.
Agricultural Planning and Risk Management: Application: Nigerian farmers face risks from unpredictable weather patterns (e.g., drought, excessive rainfall, flooding) and pest infestations. Probability helps them make decisions regarding crop selection, planting schedules, and insurance. For example, a farmer might consider the probability of sufficient rainfall in a particular month before planting a water-intensive crop.
Integration: Students can be given hypothetical scenarios involving local crops (e.g., maize, yam, cassava) and common agricultural challenges (e.g., armyworm infestation, late rains). They can use probability trees to model different outcomes (e.g., good harvest, poor harvest) based on environmental events and evaluate the likelihood of success or failure. This connects to their rural environment and the importance of agriculture to the Nigerian economy.
Sports Betting and Lottery Analysis: Application: Sports betting (especially football) is highly popular in Nigeria. While often seen as purely luck-based, understanding probability helps in evaluating the odds offered by betting companies. Similarly, national lotteries and Baba Ijebu require an understanding of permutations and combinations which form the basis of calculating winning probabilities.
Integration: Teachers can use simplified examples of sports outcomes (e.g., probability of a specific team winning, drawing, or losing based on past performance) or lottery-style number selections to illustrate how probabilities are calculated and how they inform decision-making (or highlight the low chances of winning). This taps into a culturally relevant aspect of daily life, encouraging critical thinking rather than just participation.