Probability

CIE A-Level Mathematics· Unit 4: Probability & Statistics 1, Topic 3: Probability· 20 min read

1. Core Definitions and Basic Probability★★☆☆☆⏱ 15 min

📘 Definition

Probability

$P (E)$

A measure of the likelihood of an event $E$ occurring, ranging from 0 (impossible) to 1 (certain). For equally likely outcomes, $P (E) = \frac{Number of favourable outcomes}{Total number of possible outcomes}$ .

Example:

Rolling a fair 6-sided die, the probability of rolling an even number is $\frac{3}{6} = 0.5$ .

When working with probability, the sample space describes all possible outcomes of an experiment, and an event is any subset of this sample space. A key basic concept is complementary probability: the complement of event $A$ (written $A^{'}$ ) is all outcomes not in $A$ , so $P (A^{'}) = 1 - P (A)$ .

📐 Worked Example

A bag contains 5 red, 3 blue and 2 green identical balls. A ball is drawn at random. What is the probability that the ball is not green?

1
Count the total number of balls in the bag:
2
$5 + 3 + 2 = 10$
3
Calculate the probability of drawing a green ball:
4
$P(\text{green}) = \frac{2}{10} = 0.2$
5
We want the probability of not drawing green, which is the complement of the event 'green':
6
$P(\text{not green}) = 1 - P(\text{green}) = 1 - 0.2 = 0.8$

2. Addition Rule and Mutually Exclusive Events★★☆☆☆⏱ 20 min

📘 Definition

Mutually Exclusive Events

Two events $A$ and $B$ are mutually exclusive (disjoint) if they cannot occur at the same time, meaning their intersection is empty: $A \cap B = \emptyset$ .

Example:

Drawing one card from a standard deck: getting a heart and getting a club are mutually exclusive.

The addition rule of probability calculates the probability that either $A$ , $B$ , or both occur, for any two events: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ . If $A$ and $B$ are mutually exclusive, $P (A \cap B) = 0$ , so the rule simplifies to $P (A \cup B) = P (A) + P (B)$ . Venn diagrams are a useful tool to visualize event relationships and avoid double-counting intersections.

📐 Worked Example

In a group of 50 students, 25 play hockey, 20 play football, and 10 play both sports. What is the probability a randomly selected student plays neither sport?

1
Define events: $H =$ student plays hockey, $F =$ student plays football. We know:
2
$P(H) = \frac{25}{50} = 0.5, \quad P(F) = \frac{20}{50} = 0.4, \quad P(H \cap F) = \frac{10}{50} = 0.2$
3
Use the addition rule to find the probability of playing hockey or football:
4
$P(H \cup F) = P(H) + P(F) - P(H \cap F) = 0.5 + 0.4 - 0.2 = 0.7$
5
The event of playing neither sport is the complement of $H \cup F$ :
6
$P(\text{neither}) = 1 - P(H \cup F) = 1 - 0.7 = 0.3$

✓ Quick check

Test your understanding:

For two mutually exclusive events A and B, what is the value of $P (A \cap B)$ ?
- 0
- 1
- $P (A) P (B)$
- $P (A) + P (B)$
Reveal answer
0 —
Correct! Mutually exclusive events cannot overlap, so their intersection has probability 0.

3. Multiplication Rule and Independent Events★★★☆☆⏱ 20 min

📘 Definition

Independent Events

Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other. This means $P (A ∣ B) = P (A)$ and $P (B ∣ A) = P (B)$ .

Example:

Rolling two fair dice: the outcome of the first die does not affect the outcome of the second die.

The multiplication rule states that for any two events, $P (A \cap B) = P (A ∣ B) \times P (B)$ . If $A$ and $B$ are independent, $P (A ∣ B) = P (A)$ , so the rule simplifies to $P (A \cap B) = P (A) \times P (B)$ . In CIE exams, this formula is the standard test for independence: if $P (A \cap B) = P (A) P (B)$ , then $A$ and $B$ are independent.

📐 Worked Example

A bag contains 4 red and 6 blue balls. Two balls are drawn at random with replacement. Show that the events 'first ball is red' and 'second ball is red' are independent, and find the probability both are red.

1
Define $R_{1} =$ first ball is red, $R_{2} =$ second ball is red. With replacement, the total number of balls stays the same, so:
2
$P(R_1) = \frac{4}{10} = 0.4, \quad P(R_2) = \frac{4}{10} = 0.4$
3
Calculate the probability both are red using the multiplication rule:
4
$P(R_1 \cap R_2) = P(R_1) \times P(R_2) = 0.4 \times 0.4 = 0.16$
5
Check the independence condition: $P (R_{1}) \times P (R_{2}) = 0.16 = P (R_{1} \cap R_{2})$ , so the events are independent.

4. Common Pitfalls

Wrong move:

Confusing mutually exclusive and independent events

Why:

Students often assume these terms are interchangeable, but they describe completely different properties of events

Correct move:

Remember: mutually exclusive means $P (A \cap B) = 0$ (cannot occur together), independent means $P (A \cap B) = P (A) P (B)$ (one does not affect the other). Two events with non-zero probability cannot be both.

Wrong move:

Forgetting to subtract the intersection when using the addition rule

Why:

Students use the simplified mutually exclusive formula for all events, leading to over-counting overlapping outcomes

Correct move:

Always use the general addition rule $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ unless you are explicitly told events are mutually exclusive.

Wrong move:

Multiplying unadjusted probabilities when sampling without replacement

Why:

Students assume all combined events are independent and multiply raw probabilities, ignoring the changed sample space

Correct move:

For without replacement sampling, reduce the numerator and denominator of the second probability by 1 after drawing the first item.

Wrong move:

Double-counting the intersection region in Venn diagrams

Why:

Students add the full intersection value to both individual event regions instead of only counting it once in the union

Correct move:

When filling a Venn diagram, enter the intersection value first, then subtract the intersection from each individual event to get the 'only A' and 'only B' regions.

Wrong move:

Failing to check that a probability is between 0 and 1

Why:

Rushing through calculations leads to arithmetic errors that are easily caught

Correct move:

Always verify your final probability is between 0 and 1, and re-calculate if it is outside this range.

5. Quick Reference Cheatsheet

Concept	Formula
Complement rule	$P (A^{'}) = 1 - P (A)$
General addition rule	$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Addition rule (mutually exclusive)	$P (A \cup B) = P (A) + P (B)$
General multiplication rule	$P (A \cap B) = P (A ∣ B) P (B)$
Multiplication rule (independent)	$P (A \cap B) = P (A) P (B)$
Test for independence	$P (A \cap B) = P (A) P (B)$
Mutually exclusive events	$P (A \cap B) = 0$

6. Frequently Asked

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot occur at the same time, so $P (A \cap B) = 0$ . Independent events do not affect each other's probability, so $P (A \cap B) = P (A) P (B)$ . Two events with non-zero probability cannot be both mutually exclusive and independent.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 12
Tree diagram probability calculation
2023 · 13
Venn diagram independence test
2024 · 11
Combined event probability problem

Going deeper

syllabusCIE 9709 A-Level Math SyllabusOfficial 9709 syllabus

What's Next

This foundational probability topic is required for all subsequent statistics topics in CIE A-Level 9709. After mastering basic probability rules, you will extend these concepts to conditional probability, which formalizes working with dependent events. You will also apply these core rules when studying common probability distributions like the binomial and normal distribution, which make up a large portion of the S1 exam. A solid understanding of the key rules and distinctions here will prevent common errors in almost all statistics questions you encounter.