Mathematics Form 4 | Probability | Masomotele Free Learning

📋 Learning Outcomes

By the end of this lesson, you should be able to:

Define probability and the sample space
Distinguish between experimental and theoretical probability
Apply the addition law of probability
Apply the multiplication law for independent and dependent events
Use tree diagrams and probability tables
Calculate conditional probability
Solve real-life probability problems

🔍 Key Inquiry Question: How do we measure and predict the likelihood of events in everyday life?

🎲 Basic Probability

Definition

P(A) = Number of favourable outcomes / Total outcomes

Sample Space (S): The set of all possible outcomes. E.g. for a die: S = {1,2,3,4,5,6}

Event: A subset of the sample space.

For any event A: 0 ≤ P(A) ≤ 1

P(certain event) = 1 | P(impossible event) = 0

Worked Examples

Example 1: Rolling a die

P(even) = 3/6 = 1/2 | P(prime) = 3/6 = 1/2

Example 2: Drawing a card

From a pack of 52 cards, P(King) = 4/52 = 1/13

➕ Addition Law

Mutually Exclusive Events

Events that cannot both occur at the same time.

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

(for mutually exclusive events)

Example:

P(rolling a 2 or 5) = 1/6 + 1/6 = 2/6 = 1/3

Non-Mutually Exclusive Events

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Example:

P(King or Heart) = P(K) + P(H) − P(K of Hearts) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13

Complementary Events: P(A') = 1 − P(A) | P(not A) = 1 − P(A)

✖️ Multiplication Law

Independent Events

Events where the outcome of one does not affect the other.

P(A and B) = P(A) × P(B)

Example:

Tossing a coin and rolling a die. P(Head and 6) = 1/2 × 1/6 = 1/12

Dependent Events (Conditional)

P(A and B) = P(A) × P(B|A)

P(B|A) = probability of B given A has occurred

Example:

Bag has 5 red, 3 blue balls. Two drawn without replacement.

P(both red) = 5/8 × 4/7 = 20/56 = 5/14

🌳 Tree Diagrams

Using Tree Diagrams

Tree diagrams show all possible outcomes and their probabilities.

Example: Tossing a coin twice

S = {HH, HT, TH, TT}

P(HH) = 1/2 × 1/2 = 1/4

P(at least one head) = 1 − P(TT) = 1 − 1/4 = 3/4

Example: Bag problem

Bag A: 3 red, 2 blue. Bag B: 4 red, 1 blue. Draw one from each.

P(both red) = 3/5 × 4/5 = 12/25

P(one of each) = P(RB) + P(BR) = 3/5×1/5 + 2/5×4/5 = 3/25+8/25 = 11/25

📊 Probability Tables

Two-Way Tables

Useful for organising outcomes of two events.

Example: Two dice rolled simultaneously

Die 1 \ Die 2	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9

P(sum = 7) = 6/36 = 1/6 | P(sum > 9) = 6/36 = 1/6

🔢 Conditional Probability

Definition

P(B|A) = P(A ∩ B) / P(A)

Probability of B given that A has already occurred

Example 1:

In a class: 15 boys, 10 girls. 6 boys and 4 girls play football.

P(plays football | boy) = 6/15 = 2/5

Example 2:

P(A) = 0.4, P(B) = 0.3, P(A∩B) = 0.12

P(B|A) = 0.12/0.4 = 0.3

Since P(B|A) = P(B), A and B are independent.

💡 Real-Life Applications

Medical Testing

Example:

A disease affects 1% of population. A test is 95% accurate. If a person tests positive, what is the probability they have the disease?

This uses Bayes' theorem: P(Disease|Positive) = P(Pos|Disease)×P(Disease) / P(Positive)

= (0.95×0.01) / (0.95×0.01 + 0.05×0.99) = 0.0095/0.059 ≈ 0.161 (16.1%)

Sports & Weather

Weather Example:

P(rain Mon) = 0.6, P(rain Tue | rained Mon) = 0.7

P(rain both days) = 0.6 × 0.7 = 0.42

P(rain at least one day) = 1 − P(no rain either) = 1 − (0.4×0.3) = 0.88

✏️ Practice Problems

Problem 1

A bag contains 4 red, 3 blue, 2 green balls. One is drawn at random. Find: P(red), P(not blue), P(red or green).

Solution:

Total = 9. P(red) = 4/9. P(not blue) = 6/9 = 2/3. P(red or green) = 6/9 = 2/3

Problem 2

Two cards drawn without replacement from a pack of 52. Find P(both aces).

Solution:

P(both aces) = 4/52 × 3/51 = 12/2652 = 1/221

Problem 3

P(A) = 1/3, P(B) = 1/4, A and B independent. Find P(A∪B).

Solution:

P(A∩B) = 1/3 × 1/4 = 1/12

P(A∪B) = 1/3 + 1/4 − 1/12 = 4/12 + 3/12 − 1/12 = 6/12 = 1/2

📝 Quiz — Probability

Part A: Fill in the Blanks (5 marks)

1. The probability of an impossible event is .

2. P(A') = 1 − .

3. For mutually exclusive events, P(A or B) = P(A) P(B).

4. For independent events A and B: P(A and B) = P(A) × .

5. P(rolling an even number on a die) = .

Part B: Short Answer (10 marks)

6. (4 marks) A box has 5 white, 3 black, 2 red balls. Two drawn without replacement. Find: P(both white) and P(one white, one black).

7. (3 marks) P(A)=0.5, P(B)=0.4, P(A∩B)=0.2. Find P(A∪B) and P(A|B). Are A and B independent?

8. (3 marks) A student passes Maths with probability 0.7 and English with probability 0.6. The events are independent. Find the probability of passing both, and the probability of passing at least one.