GCSE Maths Practice: listing-outcomes

Understanding Probability When Rolling a Die

This question focuses on a core GCSE Foundation concept: calculating simple probabilities when rolling a fair six-sided die. The key skill involved is identifying which outcomes meet the condition and then comparing them with the total number of possible outcomes. A standard die always has six numbers: 1, 2, 3, 4, 5, and 6. Each number has an equal chance of appearing, which makes probability calculations straightforward.

Here, the event we are interested in is rolling a 2 or a 4. These are the favourable outcomes. The total possible outcomes remain all six faces of the die. Once the favourable outcomes and total outcomes are identified, probability becomes a simple fraction calculation.

Step-by-Step Breakdown

1. First, list all numbers on the die: 1, 2, 3, 4, 5, and 6.
2. Next, identify which numbers satisfy the condition “rolling a 2 or 4”. Those numbers are 2 and 4.
3. Count the favourable outcomes. There are exactly 2 such numbers.
4. Divide the favourable outcomes by the total outcomes: 2 ÷ 6.
5. Simplify the resulting fraction if possible.

This method works not just for this specific question but for many probability questions involving identifying particular numbers or sets of numbers.

Worked Example 1: Rolling an Odd Number

The odd numbers on a die are 1, 3, and 5, which gives 3 favourable outcomes. The total remains 6. So the probability of rolling an odd number is 3/6, which simplifies to 1/2. This demonstrates how the number of favourable outcomes changes based on the condition.

Worked Example 2: Rolling a Number Greater Than 2

The numbers greater than 2 are 3, 4, 5, and 6. That gives 4 favourable outcomes. The probability is 4/6, which simplifies to 2/3. Even though the condition is different, the process—listing favourable outcomes and dividing by total outcomes—never changes.

Worked Example 3: Rolling a Number That Is a Multiple of 3

The multiples of 3 on a six-sided die are 3 and 6. So there are 2 favourable outcomes out of 6, giving a probability of 2/6 = 1/3.

Common Errors to Avoid

• Including outcomes not listed in the condition. For example, some students mistakenly include 6 because it is even, but the question only asks for 2 or 4.
• Using addition instead of division. Probability is always favourable outcomes divided by total outcomes.
• Forgetting to simplify fractions. While 2/6 is mathematically correct, simplified answers are expected in exams.
• Incorrectly assuming the die might be biased. Unless the question states otherwise, assume a die is fair.

Real-Life Connections

The logic behind dice probability extends into many real-world areas. In board games, understanding the chance of landing on certain spaces helps players make strategic decisions. In computing, random number generation relies on similar probability principles to simulate randomness. In science, experiments that involve random selection also use the same concepts to predict outcomes or test hypotheses.

FAQ

Q: Why do we divide by 6?
A: A fair die has 6 equally likely outcomes, so probability must be based on these 6 options.

Q: Do we treat 2 and 4 as separate outcomes?
A: Yes, each number counts as an individual event.

Q: Can probability ever be negative?
A: No. Probabilities always range from 0 to 1.

Study Tip

Whenever a probability question involves specific numbers on a die, highlight or mark the favourable numbers first. This simple step prevents errors and makes the calculation very clear.