Grade 10 core mathematics Statistics and Probability – Probability I (12 lessons) Notes

This 12-lesson unit introduces experimental and basic theoretical probability. Students will learn sample spaces, mutually exclusive vs independent events, the main probability laws, and tree diagrams. Lessons combine short theory, classroom experiments, guided practice, and applications in Kenyan contexts (e.g., coin/dice, drawing from a bag, school sport outcomes, weather events).

1. Identify and outline sub-sub-strands: Experimental probability; Probability space; Mutually exclusive and independent events; Laws of probability; Probability tree diagrams.
2. Perform experiments involving probabilities in different situations and collect results.
3. Identify the range of probability in different situations (0 to 1, or 0% to 100%).
4. Generate the probability space (sample space) of different events.
5. Determine the probability of mutually exclusive and independent events.
6. Apply the laws of probability (complement, addition, multiplication) in different situations.
7. Determine the probability of independent events using tree diagrams.
8. Appreciate and describe applications of probability in real-life situations.

1. Lesson 1 — Introduction to probability & experimental probability
   Objectives: Define probability; perform simple experiments (coin toss, dice) and record frequencies.

   Activities: Class coin toss (30 each), tabulate relative frequencies, discuss law of large numbers.
2. Lesson 2 — Probability as a number: range and interpretation
   Objectives: Show probability ranges from 0 to 1; convert between fraction, decimal, percentage.

   Activities: Determine probabilities of impossible (0) and certain (1) events; classroom examples (school day raining, drawing red card).
3. Lesson 3 — Sample space (probability space)
   Objectives: List sample spaces for simple experiments: coin, dice, two coins, drawing from bag.

   Activities: Create sample spaces and display on board (e.g., S for two coins = {HH, HT, TH, TT}).
4. Lesson 4 — Event notation; simple probabilities from sample spaces
   Objectives: Write events, compute P(E)= favourable/total (theoretical).

   Activities: Exercises using cards, dice and marbles in a bag.
5. Lesson 5 — Mutually exclusive events
   Objectives: Define mutually exclusive; use addition rule for mutually exclusive events: P(A or B)=P(A)+P(B).

   Activities: Roll a die: P(2 or 5); identify mutually exclusive outcomes in school contexts (pass vs fail for same student in same exam attempt).
6. Lesson 6 — Non-mutually exclusive events & union/intersection
   Objectives: Understand general addition rule: P(A∪B)=P(A)+P(B)−P(A∩B).

   Activities: Use deck/cards or class data to find overlaps (e.g., students who play football and students who study after school).
7. Lesson 7 — Complement rule and simple problem solving
   Objectives: Use P(A') = 1 − P(A) to simplify calculations.

   Activities: Compute probability of at least one head in two coin tosses by complement.
8. Lesson 8 — Independent vs dependent events (concept)
   Objectives: Define independence and dependence; test independence with P(A∩B)=P(A)P(B).

   Activities: Examples with coin tosses (independent) and drawing without replacement (dependent).
9. Lesson 9 — Multiplication rule for independent events
   Objectives: Use P(A and B)=P(A)×P(B) when independent.

   Activities: Two coin tosses, two dice; calculate exact probabilities.
10. Lesson 10 — Probability tree diagrams (independent events)
    Objectives: Draw and use tree diagrams to find combined probabilities.

    Activities: Tree for two coin tosses, tree for selecting items with replacement.
11. Lesson 11 — Applying laws & problem solving
    Objectives: Solve mixed problems combining addition, multiplication, complement and tree diagrams.

    Activities: Past KCSE-style word problems and group problem-solving.
12. Lesson 12 — Real-life applications, review & assessment
    Objectives: Describe real-life uses (weather, health risks, games, quality control) and sit a short summative assessment.

    Activities: Project presentations (mini investigations), test & feedback.

• Probability definition: For an event A, P(A) = number of favourable outcomes ÷ total number of possible outcomes (when equally likely).
• Range: 0 ≤ P(A) ≤ 1. Equivalent: 0% (impossible) to 100% (certain).
• Sample space (S): All possible outcomes. Example: Toss 1 coin S = {H, T}. Toss two coins S = {HH, HT, TH, TT}.
• Mutually exclusive: Events that cannot happen at the same time. If A and B are mutually exclusive, P(A∪B)=P(A)+P(B).
• General addition rule: P(A∪B) = P(A) + P(B) − P(A∩B).
• Complement: P(A') = 1 − P(A), where A' is the event "A does not occur".
• Independent events: A and B are independent if P(A∩B)=P(A)P(B). For independent events, P(A and B)=P(A)×P(B).
• Tree diagrams: Useful for multi-stage experiments (show branches and multiply along branches to find combined probabilities).

• Repeated coin toss (30–50 times) per group; record frequencies; compare experimental probability with theoretical.
• Dice rolling races between groups: estimate probability of sum ≥ 10 when rolling two dice.
• Marble bag: mix 5 white, 3 red, 2 blue marbles. Draw with and without replacement to show dependent vs independent events.
• Card activity: use a playing deck (or class-made cards) to practice sample spaces and intersections (suits & ranks).
• Simple survey: probability that a classmate walks to school vs uses matatu; use collected frequencies as real-life probability estimates.
• Mini-project: predict chance of rain (use local meteorological short reports) and compare with actual weather for a week.

• Exit quiz: three short questions testing complement, addition and multiplication rules.
• Practical assessment: students perform and report on an experiment (coin tosses or marble draws) and compare data with theory.
• Problem set: 6 worked questions including KCSE-style word problems.
• Group presentation: short explanation of a real-life probability application (e.g., gambling odds, weather forecasts, medical probability).

• Stronger learners: include non-uniform probability (biased coin) and conditional probability introduction.
• Supportive tasks: use physical manipulatives (coins, marbles), step-by-step guided worksheets, and pair work for weaker students.
• Extension: use simple combinations for problems with larger sample spaces (e.g., choose 2 from 5).

• Use Kenyan examples: probability of being selected in a school committee, chance of rain during exams season, results of sports teams (football) or coin toss decisions in matches.
• Discuss everyday decisions based on probability: health risks, weather planning, agriculture (chance of drought) and simple risk assessment.

• Coins, six-sided dice, coloured marbles, playing cards (or printed cards), paper and pens.
• Printed worksheets with sample space tables, tree-diagram templates, and problem sets.
• Calculator for decimal and percentage conversions.

1. Toss a coin 40 times, record heads and tails. Compute experimental probability for each. Compare to 1/2.
2. A bag contains 4 red, 3 green and 3 blue beads. Find P(red) and P(green or blue). If you draw two beads without replacement, find P(both red).
3. Draw a tree diagram for three tosses of a fair coin and find P(exactly two heads).