GCSE Maths Practice: listing-outcomes

Question 1 of 10

Practise union events by combining number properties on a six-sided die.

›

\( \begin{array}{l}\textbf{What is the probability of rolling an even} \\ \textbf{number or a number greater than 4?}\end{array} \)

Choose one option:

Remember that union counts each outcome only once.

Understanding Combined Events on a Six-Sided Die

This question involves calculating the probability of a combined event using the union of two sets. Higher-level GCSE probability often asks students to work with events such as “A or B,” which requires identifying the outcomes belonging to each set and then combining them without double-counting. The key concept is recognising overlap and understanding how union events work.

A standard six-sided die has the outcomes 1, 2, 3, 4, 5, and 6. The question asks for the probability of rolling either an even number or a number greater than 4. These conditions must be interpreted carefully to avoid counting any value twice.

Step-by-Step Reasoning

  1. List the even numbers on a die: 2, 4, 6.
  2. List the numbers greater than 4: 5, 6.
  3. Combine these sets using union. The combined set is 2, 4, 5, 6.
  4. Count the distinct outcomes: there are 4 favourable numbers.
  5. Divide by the total number of outcomes: 6.

The final probability is therefore 4/6, which simplifies to 2/3.

Worked Example 1: Rolling an Odd Number or a Multiple of 3

Odd numbers: 1, 3, 5. Multiples of 3: 3, 6. Combined union: 1, 3, 5, 6. Total favourable = 4. Probability = 4/6 = 2/3.

Worked Example 2: Rolling a Number Less Than 3 or Greater Than 5

Numbers less than 3: 1, 2. Numbers greater than 5: 6. Combined set: 1, 2, 6. Total favourable = 3. Probability = 3/6 = 1/2.

Worked Example 3: Rolling an Even Number or a Number Equal to 3

Even numbers: 2, 4, 6. Single number: 3. Combined set: 2, 3, 4, 6. Total favourable = 4. Probability = 4/6 = 2/3.

Why We Use Union (A ∪ B)

When a question contains the word “or,” it refers to the union of two sets. In probability, the formula is:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Here, the overlap is the number 6, because it is even and also greater than 4. If we simply added the counts from each set (3 even numbers + 2 numbers greater than 4), we would get 5. But 6 appears in both sets, so we would accidentally count it twice. Subtracting the overlap leaves exactly 4 unique outcomes.

Common Mistakes

  • Double-counting the overlapping value. Students often list 2, 4, 6 plus 5, 6 → incorrectly producing 5 outcomes.
  • Confusing “or” with “and.” “Or” includes all outcomes belonging to either set.
  • Forgetting to simplify fractions. While 4/6 is correct, GCSE usually expects 2/3.
  • Incorrectly assuming even numbers continue beyond 6. Limits of the die must always be checked.

Real-Life Applications

Union events appear in real decision-making, data filtering, set theory, and probability models. For example, in computing, filtering results that match condition A or condition B involves the same union concept. In risk analysis, combining different favourable scenarios uses the same logic. Understanding union-based probability builds a strong foundation for more advanced topics such as conditional probability, Venn diagrams, and probability tables.

FAQ

Q: Why does 6 appear only once?
A: Because union keeps each outcome distinct — duplicates are removed.

Q: Why not add 3/6 and 2/6?
A: Because that double-counts the overlap (number 6).

Q: Could this be solved with a Venn diagram?
A: Yes. A quick Venn diagram makes the union very clear.

Study Tip

When solving “A or B” probability questions, always list the sets separately, circle shared values, and then form the combined set without duplicates. This prevents errors and helps keep your reasoning systematic.

Related Quizzes