Introduction to Probability & Expected Frequency

Mathematics· 8.1, 8.2 (2025-2027 syllabus)· 15 min read

1. 1. Probability Scale & Key Terminology (Core)★☆☆☆☆⏱ 3 min

📘 Definition

Probability

A measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). Probabilities can also be written as percentages from 0% to 100%.

Example:

The probability of rolling a 7 on a standard 6-sided die is 0; the probability of rolling any number 1-6 is 1.

Fair: An object where every outcome has equal probability, e.g. a fair coin has a 50% chance of landing heads.
Biased: An object where outcomes do not have equal probability, e.g. a weighted coin that lands heads 70% of the time.
Random: An outcome that cannot be predicted before it occurs.

📐 Worked Example

Classify the following events as impossible, unlikely, even chance, likely, or certain: a) Rolling a number less than 7 on a standard 6-sided die, b) Winning a raffle with 1 ticket out of 1000 total tickets, c) Getting heads when flipping a fair coin.

1
For part a: All possible outcomes of a die roll (1-6) are less than 7, so this event is certain.
2
For part b: Only 1 out of 1000 tickets wins, so this event is unlikely.
3
For part c: A fair coin has equal chance of landing heads or tails, so this is an even chance event.

2. 2. Single Event Probability & Its Complement★★☆☆☆⏱ 4 min

The probability of a single event occurring is calculated as the number of favourable outcomes divided by the total number of equally likely possible outcomes:

P(\text{event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of equally likely outcomes}}

📘 Definition

Complement of an Event

P(A') (Extended only)

The probability that an event does NOT occur, equal to 1 minus the probability of the event occurring.

Example:

If the probability of rain tomorrow is 0.3, the probability of no rain is $1 - 0.3 = 0.7$ . Extended students write this as $P (rain^{'}) = 0.7$ .

📐 Worked Example

A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. All marbles are equally likely to be picked. a) Calculate the probability of picking a red marble. b) Use the complement rule to calculate the probability of NOT picking a blue marble.

1
First calculate total number of marbles: $3 + 5 + 2 = 10$ .
2
Part a: Number of favourable outcomes (red marbles) = 3, so $P (red) = \frac{3}{10} = 0.3$ .
3
Part b: First find $P (blue) = \frac{5}{10} = 0.5$ . The complement (not blue) is $1 - 0.5 = 0.5$ .

✓ Quick check

If the probability of a football team winning a match is 0.45, what is the probability they do not win?
Reveal answer
0.55 —
Use the complement rule: $1 - 0.45 = 0.55$ .

3. 3. Relative Frequency (Core)★★☆☆☆⏱ 4 min

When you do not know the theoretical probability of an event (for example, for a biased object), you can estimate it using relative frequency, calculated from repeated trial data:

\text{Relative frequency} = \frac{\text{Number of times event occurs}}{\text{Total number of trials}}

📐 Worked Example

A biased 6-sided die is rolled 200 times. The number 4 comes up 60 times. Estimate the probability of rolling a 4 on this die.

1
Number of times the event (rolling a 4) occurs = 60. Total number of trials = 200.
2
Relative frequency = $\frac{60}{200} = 0.3$ . This is your estimate of the probability of rolling a 4 on the biased die.

4. 4. Expected Frequency (Core)★★★☆☆⏱ 4 min

Expected frequency is the number of times you expect an event to occur over a given number of trials, calculated by multiplying the event's probability by the total number of trials:

\text{Expected frequency} = P(\text{event}) \times \text{Number of trials}

📐 Worked Example

The probability of winning a game at a fair is 0.12. If 250 people play the game, how many people are expected to win?

1
Identify values: $P (w in) = 0.12$ , number of trials = 250.
2
Calculate expected frequency: $0.12 \times 250 = 30$ . 30 people are expected to win.

5. Extended Only: Formal Probability Notation★★☆☆☆Extended only⏱ 2 min

📘 Definition

Probability Notation

$P (A)$ , $P (A^{'})$

$P (A)$ = probability of event A occurring. $P (A^{'})$ = probability of the complement of A (event A not occurring).

Example:

If $P (passing an exam) = 0.8$ , then $P (failing the exam) = P (pass^{'}) = 1 - 0.8 = 0.2$ .

📐 Worked Example

Event A is 'drawing a heart from a standard deck of 52 cards'. a) Write $P (A)$ as a simplified fraction. b) Write $P (A^{'})$ as a simplified fraction.

1
Number of hearts in a deck = 13, total cards = 52.
2
Part a: $P (A) = \frac{13}{52} = \frac{1}{4}$ .
3
Part b: $P (A^{'}) = 1 - \frac{1}{4} = \frac{3}{4}$ .

6. Common Pitfalls

Wrong move:

Calculating probability as total outcomes divided by favourable outcomes

Why:

You have inverted the fraction, leading to a value greater than 1 which is mathematically invalid for probability.

Correct move:

Always divide the number of favourable outcomes by the total number of equally likely outcomes.

Wrong move:

Assuming all outcomes are equally likely for biased objects

Why:

Biased objects do not have equal probability per outcome, so theoretical probability calculations assuming equal likelihood are incorrect.

Correct move:

Use relative frequency from trial data to estimate probability for biased objects.

Wrong move:

Treating expected frequency as a guaranteed exact number of occurrences

Why:

Expected frequency is a statistical estimate, and real trial results will naturally vary around this value.

Correct move:

State expected frequency as an estimate, and recognize that real results may differ from the calculated value.

Wrong move:

Using $P (A)$ to represent the complement of event A (Extended only)

Why:

$P (A)$ explicitly refers to the probability of event A occurring, not its opposite.

Correct move:

Use the $P (A^{'})$ notation to represent the complement of event A.

Wrong move:

Submitting probability values less than 0 or greater than 1

Why:

Probability is only defined between 0 (impossible) and 1 (certain). Values outside this range indicate a calculation error.

Correct move:

Double check your working immediately if your result falls outside the 0 to 1 range.

7. Quick Reference Cheatsheet

Concept	Core Rule/Formula	Extended Notation
Probability Scale	All values between 0 (impossible) and 1 (certain)
Single Event Probability	Favourable outcomes / Total equally likely outcomes	$P (A) = \frac{n ( A )}{n ( T )}$
Complement of Event	1 - Probability of the event	$P (A^{'}) = 1 - P (A)$
Relative Frequency	Event occurrences / Total number of trials
Expected Frequency	Probability of event × Number of trials	Expected frequency = $P (A) \times n$

8. Frequently Asked

Do I use fractions or decimals for probability answers?

Either is acceptable unless the question specifies, but simplified fractions are preferred for exact values. Always round decimals to 2-3 significant figures if no guidance is given.

What is the difference between relative frequency and probability?

Probability is the theoretical chance of an event occurring, while relative frequency is a real-world estimate of probability calculated from data collected during repeated trials.

What's Next

Now that you have mastered foundational probability concepts for CIE IGCSE Maths 0580, you are ready to move on to combined probability events, the next sub-topic in the Probability unit. You will apply the core rules you have learned (probability calculation, complement rule, expected frequency) to problems involving two or more events, including independent and dependent events, mutually exclusive events, and probability trees. These skills are frequently tested in both Core and Extended tier papers, often as part of longer structured questions worth 4-6 marks. Make sure you are comfortable simplifying fractions and working with decimals before moving on, as these skills are essential for all probability calculations.