GRADE 9 Mathematics DATA HANDLING AND PROBABILITY – PROBABILITY Notes
Mathematics — Data Handling & Probability
Subtopic: PROBABILITY (Age ~14 — Kenyan context)
Learning objectives
- Understand what probability means and the probability scale (0 to 1).
- Find probability using P(Event) = favourable outcomes / total outcomes.
- Differentiate theoretical and experimental probability.
- Use simple rules: complements, union (or), intersection (and), independence.
- Outcome: a single possible result (e.g., getting Heads when tossing a coin).
- Sample space (S): all possible outcomes (e.g., S for a coin = {H, T}).
- Event: one or more outcomes we are interested in (e.g., getting an even number).
- Favourable outcomes: outcomes that make the event happen.
- Probability: a number between 0 and 1 showing how likely an event is.
0 = impossible, 1 = certain. Values between show chance. Example: 0.5 (1/2) = even chance.
P(Event) = number of favourable outcomes / total number of possible equally likely outcomes.
Always simplify fractions when possible.
- 0 ≤ P(E) ≤ 1
- Complement: P(E') = 1 − P(E) where E' is "not E".
- Mutually exclusive events (cannot happen together): P(A or B) = P(A) + P(B).
- General addition: P(A or B) = P(A) + P(B) − P(A and B).
- Independent events (one does not affect the other): P(A and B) = P(A) × P(B).
Example 1 — Coin
If you toss a fair coin once, what is P(Heads)?
Sample space S = {H, T} so total = 2. Favourable = {H} → 1.
P(Heads) = 1/2 = 0.5.
Example 2 — Dice
Roll one fair six-sided die. Find P(roll a 4) and P(roll an even number).
Total outcomes = 6 (1–6). Favourable for 4 = 1 → P = 1/6.
Favourable for even = {2,4,6} → 3 outcomes → P = 3/6 = 1/2.
Example 3 — Marbles (without replacement)
A bag has 5 red, 3 blue and 2 green marbles (total 10). Two marbles are taken one after the other without replacement. What is the probability both are blue?
First draw: P(first blue) = 3/10.
After one blue taken: left blue = 2, total = 9, so P(second blue | first blue) = 2/9.
For both: multiply (dependent events) → (3/10) × (2/9) = 6/90 = 1/15 ≈ 0.0667.
Example 4 — Using union and intersection
Roll a die. Find P(event A = "even") and P(event B = "multiple of 3"), then P(A or B).
A = {2,4,6} → P(A)=3/6=1/2. B = {3,6} → P(B)=2/6=1/3.
A and B = {6} → P(A and B)=1/6.
Use P(A or B) = P(A)+P(B)−P(A and B) = 1/2 + 1/3 − 1/6 = (3+2−1)/6 = 4/6 = 2/3.
Steps:
- Toss a coin 30 times and record Heads/Tails.
- Experimental P(Heads) = number of Heads / 30.
- Compare this number to theoretical 1/2. Discuss differences — more trials give closer results.
- One fair die is rolled. Find P(rolling an odd number).
- A bag contains 4 white and 6 black balls. One ball is taken at random. Find P(white).
- Toss two coins. What is the probability of getting exactly one head?
- From a pack of 10 cards numbered 1 to 10, one card is drawn. Find P(the number is a multiple of 3).
- A spinner is divided into 5 equal parts labelled A, B, C, D, E. If you spin once, what is P(A or B)?
Answers
- Odd numbers = {1,3,5} → 3/6 = 1/2.
- P(white) = 4/(4+6) = 4/10 = 2/5.
- Two coins sample space = {HH, HT, TH, TT}. Exactly one head = {HT, TH} → 2/4 = 1/2.
- Multiples of 3 from 1–10: {3,6,9} → 3/10.
- Two favourable parts out of 5 → 2/5.
- Always write the sample space first if unsure.
- Remember to count equally likely outcomes only when using the simple formula.
- When drawing without replacement, the second probability usually changes (dependent events).
- For "or" questions, check whether events can happen together — do not just add probabilities if they overlap.
Summary: Probability tells how likely an event is. Use P = favourable / total, remember the scale 0–1, and apply complement, addition and multiplication rules for more complex questions.