Essential Mathematics — Statistics & Probability

Subtopic: Probability I (Age 15 — Kenyan context)

This note introduces basic ideas of probability for single- and two-step experiments. It covers equally likely outcomes, how to compute probability, the range of probability, mutually exclusive events, independent events, and simple experiments you can do in class or at home. Aim: by the end, learners will be able to perform small experiments and calculate probabilities in everyday situations.

1. What is probability?

Probability measures how likely an event is to happen. For a simple experiment with equally likely outcomes:

Probability of event A = (number of favourable outcomes) / (total number of possible outcomes)

2. Equally likely outcomes

Outcomes are equally likely when each outcome has the same chance of occurring. Examples:

• Tossing a fair coin: {Heads, Tails} — both equally likely.
• Rolling a fair six-sided die: {1,2,3,4,5,6} — all equally likely.
• Picking a ball from a bag when all balls look identical except colour, and mixing is thorough.

3. Range of probability

Probability is always between 0 and 1 (inclusive):

• P = 0 means the event is impossible (will not happen).
• P = 1 means the event is certain (will always happen).
• 0 < P < 1 means the event may or may not happen.

4. Mutually exclusive events

Two events are mutually exclusive (disjoint) if they cannot happen at the same time.

• Example (die): getting an odd number and getting an even number are mutually exclusive.
• Example (coin): Heads and Tails on one toss are mutually exclusive.

For mutually exclusive events A and B:

P(A or B) = P(A) + P(B)

5. Independent events

Two events are independent when the outcome of one does not affect the outcome of the other.

• Example: tossing a coin and rolling a die (on separate devices) — the coin result does not change the die result.
• Selecting with replacement: pick a marble, note its colour, replace it, then pick again → selections independent.

For independent events A and B:

P(A and B) = P(A) × P(B)

6. Single-chance experiments

A single-chance experiment is one trial where we record one outcome. Examples for class experiments:

• Toss one coin once, record Heads or Tails.
• Roll one die once, record the face (1–6).
• Pick one sweet from a bag without looking and note its colour.

Use the formula P = favourable / total when outcomes are equally likely.

7. Practical examples and worked solutions

8. Classroom experiments and activities (Suggested Learning Experiences)

These activities are low-cost and use items commonly available in Kenyan schools and homes.

1. Toss-and-record (Equally likely outcomes)
   • Materials: coin (or bottle for spinning), table, notebook.
   • Procedure: Each learner tosses the coin 50 times, records Heads/Tails count, computes relative frequency Heads = (Heads count)/50. Compare class values to theoretical P(Heads)=0.5. Discuss variation and law of large numbers.
2. Marble experiment (without/with replacement)
   • Materials: bag with 5 red, 3 black, 2 white stones or sweets.
   • Procedure A (without replacement): draw one, do not replace, draw second. Record colours — show how second draw probabilities change (dependent events).
   • Procedure B (with replacement): draw one, replace it, draw again. Show independence and calculate P(both red) = P(red)×P(red).
3. Two-dice sums (Independent events)
   • Materials: two dice per pair of learners.
   • Procedure: Roll both dice many times, record sums. Calculate experimental probability for sum = 7 and compare to theoretical 6/36 = 1/6. Discuss why some sums are more likely.
4. School context examples
   • Pick a random student from a class and record their birth month. Estimate P(born in March) experimentally and compare with assumption of equal months (1/12). Discuss seasonal birth patterns in the community.
5. Real-life discussion (appreciation)
   • Discuss where probability is used: weather forecasts (chance of rain), epidemiology (chance of infection), quality control, and making decisions under uncertainty.

9. Short exercises (with answers)

1. One fair die is rolled. Find P(odd number). Answer: odd faces {1,3,5} → 3/6 = 1/2.
2. From a pack of 52 cards (standard), pick 1 card. Find P(a king). Answer: 4/52 = 1/13.
3. A bag has 4 mangoes and 6 oranges. Pick one fruit at random. P(mango) = 4/10 = 2/5 = 0.4.
4. Two fair coins are tossed. Find P(both Heads). Answer: (1/2)×(1/2)=1/4.
5. Are the events “drawing a red sweet” and “drawing a blue sweet” from a single draw mutually exclusive? Answer: Yes — cannot be both on the same single draw.

10. Key formulas & reminders

• P(A) = favourable outcomes / total outcomes (for equally likely outcomes).
• 0 ≤ P(A) ≤ 1. Also useful forms: as fraction, decimal, or percentage.
• If A and B are mutually exclusive: P(A or B) = P(A) + P(B).
• If A and B are independent: P(A and B) = P(A) × P(B).
• For two events (not mutually exclusive): P(A or B) = P(A) + P(B) − P(A and B).

11. Assessment tasks for the teacher

Give students one or two of the following for classwork or short test:

• Design and carry out a 30-trial experiment to estimate the probability of rolling a multiple of 3 on a die. Compare experimental and theoretical results.
• Explain with an example the difference between mutually exclusive and independent events.
• A jar contains 7 red, 5 green and 8 yellow beads. One bead is drawn at random. Calculate the probability it is not green.

12. Final notes (curriculum links & life skills)

The ideas here link with Statistics & Probability topics in Kenyan secondary school syllabi. Emphasize practical experiments, recording data, and interpreting results. Teach learners to think critically about probability statements in newspapers and weather reports, and to understand that probability gives a measure of uncertainty, not certainty.

Prepared for Essential Mathematics — Probability I. Activities are safe and inexpensive; adapt examples to local materials (stones, sweets, paper cards, coins).