GCSE Maths Practice: probability-basics

Understanding Probability with Multiple Conditions

This question is a classic higher-tier GCSE Maths probability problem. It requires working with two layers of conditions: the card must be red and the card must not be a face card. These multi-condition questions help build advanced reasoning skills, especially relevant in set-based probability, overlapping groups and complementary events.

Red Cards in a Standard Deck

A standard deck contains 52 cards. Half of these—26 cards—are red. The red suits are:

• Hearts (13 cards)
• Diamonds (13 cards)

Each suit contains the same ranks: numbers 2–10, plus Jack, Queen and King. Understanding this symmetry is essential for quickly calculating probabilities without recounting every card.

Face Cards

Face cards are:

• Jack
• Queen
• King

Each suit has one of each, meaning 3 face cards per suit. Since there are two red suits, the number of red face cards is:

3 × 2 = 6 red face cards

These cards must be excluded because the problem asks for red cards that are not face cards.

Counting Favourable Outcomes

Total red cards: 26
Total red face cards: 6
Non-face red cards: 26 − 6 = 20

These 20 cards form the favourable outcomes for the event.

Worked Example 1: Black Non-Face Cards

Black cards (clubs and spades) also total 26. Each black suit also contains 3 face cards, giving 6 black face cards. Therefore, black non-face cards = 26 − 6 = 20. This parallel structure reinforces card symmetry in GCSE probability.

Worked Example 2: Red Number Cards Only

If the condition changes to drawing a red number card (2–10), each suit has 9 number cards. Two red suits → 18 number cards. Probability would be 18/52. This variation helps build fluency in categorising card subsets.

Worked Example 3: Non-Face Cards of Any Colour

Total face cards = 12 (3 per suit × 4 suits). Non-face cards = 52 − 12 = 40. This demonstrates how complementary methods quickly identify card groups.

Common Mistakes

• Thinking Aces are face cards: Aces are not face cards.
• Forgetting to subtract face cards: Many learners count all red cards without removing the 6 face cards.
• Mixing independent and combined events: This problem requires counting—multiplying probabilities is not appropriate here.

Real-Life Applications

The ability to categorise overlapping groups is essential for probability in real-world contexts such as statistics, machine learning, quality control, and risk modelling. Many real systems involve events that meet several conditions at once—just like a card must be red and non-face. This type of reasoning forms the foundation for more advanced A-level topics like set notation and conditional probability.

FAQ

Q: Are Aces included in non-face cards?
Yes. Aces are not considered face cards.

Q: Could the fraction be simplified?
It can be simplified to 5/13, but unless the question asks for simplest form, 20/52 is acceptable.

Q: Do red suits always have the same number of face cards?
Yes. Each suit contains exactly one Jack, one Queen and one King.

Study Tip

Whenever working with playing cards in higher-tier GCSE questions, separate the deck into predictable structures (suits, face cards, number cards). This helps avoid overcounting and makes set-based probability much easier to solve.