GCSE Maths Practice: probability-basics

Understanding Probability with Prime Numbers

When working with GCSE Higher probability, questions often involve numerical properties such as factors, multiples, and prime numbers. A prime number is a positive integer greater than 1 that has exactly two factors: 1 and itself. On a standard fair 6-sided die, the numbers are 1, 2, 3, 4, 5, and 6. Identifying prime numbers requires understanding basic number theory, which makes this type of question more challenging than simple even-or-odd probability.

The prime numbers between 1 and 6 are 2, 3, and 5. The number 1 is not prime because it has only one factor. The numbers 4 and 6 are composite because they have more than two factors. Once the prime numbers are identified, probability is calculated by dividing the number of favourable outcomes by the total number of possible outcomes, which is always 6 for a fair die.

Step-by-Step Method

1. List all the numbers on the die.
2. Identify which numbers satisfy the condition (in this case, being prime).
3. Count how many favourable outcomes there are.
4. Divide the favourable outcomes by the total number of outcomes (6).

Worked Example 1

Find the probability of rolling a factor of 6. The factors of 6 are 1, 2, 3, and 6. There are 4 favourable outcomes out of 6, so the probability is \(\frac{4}{6} = \frac{2}{3}\). Notice how this uses number properties in a similar way to identifying prime numbers.

Worked Example 2

Find the probability of rolling a multiple of 2. The multiples of 2 on a die are 2, 4, and 6. That gives 3 favourable outcomes, so the probability is \(\frac{3}{6} = \frac{1}{2}\). This acts as a structural parallel to the prime-number question but uses a different property.

Common Mistakes

• Students sometimes think 1 is prime. It is not.
• Some forget that probability must always be a value between 0 and 1.
• Others mix up factors, multiples, and primes, leading to incorrect favourable outcomes.
• Another common error is simplifying fractions incorrectly.

Real-Life Applications

Prime-number reasoning is essential not only in dice questions but also in encryption, coding, error detection, and data structures. Understanding numerical properties strengthens logical skills used across mathematics, science, and computer science, including GCSE and A-level content.

FAQ

Q: Why is 1 not prime?
Because prime numbers must have exactly two factors.

Q: Can a die have more than one prime number?
Yes, depending on the faces. A standard 6-sided die has three primes.

Q: Do I always list outcomes?
For Higher-tier reasoning, listing helps avoid mistakes, especially with number-property questions.

Study Tip

Memorise the prime numbers up to 20. It speeds up many GCSE Higher probability questions and improves fluency in number reasoning.