Probability: single event probability, expected frequency, relative frequency

What you'll learn

Probability forms a substantial component of the CIE IGCSE Mathematics syllabus, appearing consistently across both Extended and Core tier papers. This guide covers single event probability calculations, expected frequency predictions, and relative frequency from experimental data—skills that regularly appear in Paper 2 and Paper 4 questions worth 3-5 marks each.

Key terms and definitions

Probability — A numerical measure between 0 and 1 (or 0% to 100%) indicating how likely an event is to occur, where 0 represents impossibility and 1 represents certainty.

Single event probability — The likelihood of one specific outcome occurring from a single trial or experiment, calculated as the number of favourable outcomes divided by the total number of possible outcomes.

Expected frequency — The predicted number of times an outcome will occur over a given number of trials, calculated by multiplying the probability of the event by the total number of trials.

Relative frequency — An estimate of probability based on experimental data, calculated as the number of times an event occurred divided by the total number of trials conducted.

Sample space — The set of all possible outcomes in a probability experiment.

Outcome — A single possible result from a probability experiment.

Mutually exclusive events — Events that cannot occur simultaneously; if one happens, the other cannot.

Theoretical probability — The calculated probability based on equally likely outcomes, assuming perfect conditions without experimental testing.

Core concepts

Calculating single event probability

The fundamental probability formula for a single event is:

P(event) = Number of favourable outcomes / Total number of possible outcomes

For this formula to work accurately, all outcomes must be equally likely. This assumption underpins most CIE IGCSE probability questions.

When working with single event probability:

• Identify the total number of possible outcomes in the sample space
• Count how many of those outcomes satisfy the condition you're investigating
• Express the probability as a fraction in its simplest form (CIE mark schemes often require this)
• Convert to decimals or percentages only when the question explicitly requests it

Example contexts from CIE papers:

• Drawing a card from a standard 52-card deck
• Rolling a fair six-sided die
• Selecting a coloured ball from a bag
• Spinning a fair spinner with numbered sections
• Choosing a student from a class with given characteristics

The probability scale ranges from 0 (impossible) to 1 (certain). Events with probability 0.5 are equally likely to happen or not happen.

Understanding and applying expected frequency

Expected frequency connects theoretical probability with real-world predictions. The formula is:

Expected frequency = Probability × Number of trials

This calculation tells you how many times you would expect an outcome to occur if an experiment were repeated a specific number of times.

Key points for CIE IGCSE examinations:

• Expected frequency can be a decimal or whole number—do not round unless instructed
• The question typically provides both the probability (or information to calculate it) and the number of trials
• Answers should match the context: "approximately 150 people" or "about 75 times"
• Expected frequency represents a long-term average, not a guarantee

When answering expected frequency questions:

1. Calculate or identify the single event probability first
2. Identify the total number of trials clearly stated in the question
3. Multiply these values
4. Present your answer with appropriate units or context from the question

Expected frequency questions often appear in real-world contexts: quality control in manufacturing, weather predictions over a year, or estimating attendance at events.

Working with relative frequency

Relative frequency provides an experimental approach to estimating probability. Unlike theoretical probability, relative frequency uses actual data collected from trials.

The formula is:

Relative frequency = Number of times event occurred / Total number of trials

Relative frequency is particularly valuable when:

• Theoretical probability is difficult or impossible to calculate
• Outcomes are not equally likely
• Real experimental data is available
• Testing whether a dice, spinner, or coin is fair

Important characteristics of relative frequency for CIE examinations:

• As the number of trials increases, relative frequency typically becomes a better estimate of theoretical probability (the law of large numbers)
• Relative frequency can differ from theoretical probability due to experimental variation
• Express relative frequency as a decimal or fraction as required
• Relative frequency always falls between 0 and 1

CIE IGCSE questions frequently present a table of experimental results and ask students to:

• Calculate relative frequencies for different outcomes
• Compare relative frequency with theoretical probability
• Determine whether equipment (dice, coins) appears to be biased
• Predict future outcomes based on experimental data

Converting between probability representations

CIE IGCSE Mathematics requires fluency in expressing probability as:

• Fractions (most common in mark schemes): e.g., 3/8
• Decimals: e.g., 0.375
• Percentages: e.g., 37.5%

Conversion skills tested:

• Simplifying fractions to lowest terms (3/12 becomes 1/4)
• Converting fractions to decimals by division
• Converting decimals to percentages by multiplying by 100
• Working backwards from percentages to fractions

The question wording indicates the required form: "Give your answer as a fraction" or "Write your answer as a percentage."

Complementary events

The probability that an event does not occur is found using:

P(not A) = 1 − P(A)

This relationship appears regularly in CIE papers, often as part of multi-step problems.

For example:

• If P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7
• If P(passing) = 7/10, then P(not passing) = 1 − 7/10 = 3/10

Understanding complementary events helps when:

• The probability of an event not happening is easier to calculate
• Checking answers (probabilities of all outcomes in a sample space sum to 1)
• Solving problems involving "at least one" scenarios in Extended tier

Worked examples

Example 1: Single event probability

Question: A box contains 5 red balls, 3 blue balls, and 2 green balls. One ball is selected at random. Calculate the probability that the ball is: (a) blue (b) not green

Solution:

(a) Total number of balls = 5 + 3 + 2 = 10

Number of blue balls = 3

P(blue) = 3/10 ✓

(b) Number of balls that are not green = 5 + 3 = 8

P(not green) = 8/10 = 4/5 ✓

Alternative method for (b): P(green) = 2/10 = 1/5 P(not green) = 1 − 1/5 = 4/5 ✓

Marks: (a) 1 mark for correct probability in simplest form; (b) 1 mark for correct probability

Example 2: Expected frequency

Question: The probability that a machine produces a defective component is 0.04. In one day, the machine produces 850 components. How many defective components would you expect?

Solution:

Expected number of defective components = Probability × Total number

= 0.04 × 850 ✓

= 34 ✓

You would expect 34 defective components.

Marks: 1 mark for correct method; 1 mark for correct answer (2 marks total)

Example 3: Relative frequency

Question: A student throws a drawing pin 200 times and records whether it lands point up or point down.

(a) Calculate the relative frequency of the pin landing point up. (b) The student throws the pin another 50 times. Estimate how many times it will land point up.

Solution:

(a) Relative frequency of point up = 132/200 ✓

= 0.66 (or 33/50 or 66%) ✓

(b) Using relative frequency as an estimate of probability:

Expected frequency = 0.66 × 50 ✓

= 33 ✓

Estimate: 33 times point up

Marks: (a) 2 marks (1 for method, 1 for answer); (b) 2 marks (1 for using relative frequency, 1 for correct calculation)

Common mistakes and how to avoid them

• Mistake: Forgetting to simplify fractions in probability answers. Writing 6/12 instead of 1/2 may lose marks if the question specifies "in its simplest form" or mark schemes indicate simplified answers only. Correction: Always check if your fraction can be simplified by finding common factors of numerator and denominator.

• Mistake: Confusing probability and expected frequency. Writing P(heads) = 50 when flipping a coin 100 times, rather than stating this is the expected frequency. Correction: Probability is always between 0 and 1. Expected frequency is probability multiplied by number of trials and represents a count of occurrences.

• Mistake: Adding probabilities incorrectly when calculating "not" probabilities. For example, calculating P(not 6) on a die as 1/6 + 1/6 + 1/6 + 1/6 + 1/6 instead of using the complement. Correction: Use P(not A) = 1 − P(A). For the die example: P(not 6) = 1 − 1/6 = 5/6.

• Mistake: Treating relative frequency as exact probability. Writing "the probability is definitely 0.66" when only 200 trials have been conducted. Correction: Relative frequency provides an estimate that improves with more trials. Use phrases like "the estimated probability is..." or "based on these results..."

• Mistake: Calculating probability with the formula inverted: using total outcomes divided by favourable outcomes. Correction: Always write P(event) = favourable/total. The smaller number (favourable outcomes) goes on top; the larger number (total outcomes) goes on the bottom.

• Mistake: Rounding expected frequency to whole numbers when the question doesn't require it. Writing "approximately 34" when the answer is 33.6. Correction: Keep expected frequency as a decimal unless the context requires whole numbers (e.g., "number of people") or the question explicitly asks for rounding.

Exam technique for Probability: single event probability, expected frequency, relative frequency

• Command words matter: "Calculate" requires a numerical answer with working shown. "Estimate" (often with relative frequency) allows use of experimental data as your probability value. "Find" requires the probability value, usually as a fraction in simplest form. Show all working for method marks even if your final answer is incorrect.

• Structure for multi-part questions: CIE probability questions often build in difficulty: part (a) asks for a simple probability, part (b) asks for expected frequency using that probability, part (c) might ask for relative frequency from new data. Complete each part methodically and use previous answers in subsequent parts when appropriate.

• Mark allocation guides working: A 1-mark question typically requires only a final answer (often simple probability). A 2-mark question needs method shown: one mark for correct approach, one for correct answer. A 3-mark question might involve multiple steps: calculate probability (1 mark), then expected frequency (1 mark for method, 1 for answer).

• Precision with fractions: Extended tier mark schemes frequently specify fractions in lowest terms. Core tier may accept equivalent fractions, but simplifying never loses marks and demonstrates mathematical rigour. When converting between forms, maintain accuracy: use exact fractions where possible before converting to decimals.

Quick revision summary

Single event probability equals favourable outcomes divided by total outcomes, expressed as a value between 0 and 1. Expected frequency predicts how many times an event occurs over repeated trials by multiplying probability by number of trials. Relative frequency estimates probability from experimental results by dividing observed occurrences by total trials conducted. Complementary probability uses P(not A) = 1 − P(A). Always simplify fractions, show working clearly, and match your answer format to question requirements for full marks in CIE IGCSE Mathematics examinations.