GCSE Maths Practice: probability-basics

Understanding Probability Using Even Numbers on a Die

Probability questions involving dice are a core part of GCSE Maths because the outcomes are simple, fixed and equally likely. A fair six-sided die has faces numbered 1 to 6, and each face has the same chance of appearing. This makes dice ideal for practising basic calculations such as identifying favourable outcomes and applying the probability formula.

What Counts as an Even Number?

An even number is a number divisible by 2. On a six-sided die, the even numbers are 2, 4 and 6. Since there are three such numbers, you immediately know that there are three favourable outcomes for this event.

Total Possible Outcomes

The die has six faces, so there are six possible outcomes each time you roll it. This total does not change unless you are using a different type of die or more than one die. In GCSE questions, a standard die is always assumed unless stated otherwise.

The Probability Formula

The rule for calculating probability is:

Probability = (Number of favourable outcomes) ÷ (Total number of outcomes)

For this question, the favourable outcomes are 2, 4 and 6, giving three outcomes. Dividing 3 by 6 gives the probability.

Worked Example 1: Rolling an Odd Number

The odd numbers on a die are 1, 3 and 5. That gives three favourable outcomes. Using the formula gives 3/6, which simplifies to 1/2. This mirrors the reasoning used for even numbers and highlights the symmetry in the number set.

Worked Example 2: Rolling a Multiple of 3

The multiples of 3 on a die are 3 and 6, giving two favourable outcomes. The probability would therefore be 2/6, which simplifies to 1/3. This example helps reinforce the process of identifying specific number properties.

Worked Example 3: Rolling a Number Less Than 5

The numbers less than 5 are 1, 2, 3 and 4. That makes four favourable outcomes. Using the formula gives 4/6, simplifying to 2/3. This shows how the same method applies whether the condition is simple or slightly broader.

Common Mistakes

• Including numbers that do not match the condition, such as counting 1 by mistake.
• Confusing “even” with “multiple of 2” but forgetting the full list.
• Thinking that one outcome becomes more likely because of previous rolls.

Real-Life Applications

Dice probability mirrors many real-life ideas where outcomes are equally likely. These concepts appear in board games, probability simulations, coding random number generators and analysing risk in mathematics. A strong understanding of dice probability also supports more advanced GCSE topics including independent events, tree diagrams and compound probability.

FAQ

Q: Does rolling the die multiple times change the probability?
No. Each roll is independent. Past results do not affect the next roll.

Q: Should 3/6 be simplified?
You may simplify it to 1/2 if the question asks for simplest form, but both forms are valid unless stated otherwise.

Q: What if the die is not fair?
GCSE questions always assume a fair die unless the question explicitly mentions bias.

Study Tip

When working with dice, always list all six outcomes first: 1, 2, 3, 4, 5, 6. Then highlight only the numbers that satisfy the condition. This avoids missing or adding unintended values and ensures accurate probability calculations every time.