GCSE Maths Practice: listing-outcomes

Understanding Probability with Face Cards in a Standard Deck

This question explores a key Higher GCSE probability skill: classifying items in a set and calculating probabilities using structured counting. A standard deck contains 52 distinct cards made up of four suits: Hearts, Diamonds, Clubs, and Spades. Each suit contains 13 ranks, including three face cards: the Jack, Queen, and King. This gives a total of 12 face cards in the entire deck. Since each of these 12 cards is equally likely to be drawn when one card is picked at random, the probability is found by comparing favourable outcomes (face cards) with the total number of possibilities (all cards in the deck).

In more complex probability problems, you are asked to analyse subsets of a larger sample space. This question is an example of that type of reasoning. Students must recognise which cards qualify as face cards, count them accurately, and then express the probability as a fraction. Although the calculation is not difficult, the classification step requires careful attention, which makes it suitable for Higher level.

Step-by-Step Method

1. Recall the structure of a 52-card deck.
2. Identify the ranks that count as face cards: Jack, Queen, and King.
3. Multiply the number of face cards per suit (3) by the number of suits (4).
4. Determine the total number of favourable outcomes: 12.
5. Form the probability fraction: 12 ÷ 52.
6. Simplify if needed (the fraction simplifies to 3/13, though many exam questions accept 12/52).

These steps demonstrate how probability can be broken down into logic, counting, and proportional comparison.

Worked Example 1: Probability of Drawing a Queen

There are 4 Queens in the deck. Therefore, the probability is 4/52, which simplifies to 1/13. This uses the same technique: count the relevant cards and divide by the total.

Worked Example 2: Probability of Drawing a Heart

There are 13 Hearts. The probability is 13/52 = 1/4. Identifying suits as categories helps apply the same reasoning used in counting face cards.

Worked Example 3: Probability of Drawing Any Picture Card

If a question includes Jacks, Queens, Kings, and Aces (treating Aces as special cards), there would be 16 picture cards. The method remains the same: count the favourable outcomes, then divide by 52.

Common Mistakes

• Including Aces as face cards. In mathematics and card theory, only Jacks, Queens, and Kings are counted as face cards. Aces are separate.
• Thinking probabilities must already be simplified. Both 12/52 and 3/13 are mathematically correct, but simplification may be expected.
• Assuming Jokers are included. Standard probability problems use decks without Jokers unless stated.
• Forgetting that each suit is identical in structure. Every suit has exactly 3 face cards.

Real-Life Applications

Card-based probability problems arise in gaming, statistics, computing, and simulations. For example, probability models in card games rely on understanding how many cards satisfy certain conditions. In computing, random simulations use similar logic when modelling selective outcomes. In data science, categorising elements of a dataset into types mirrors the process used here to classify cards.

FAQ

Q: Why are there 12 face cards?
A: Each of the four suits contains a Jack, Queen, and King, giving 3 × 4 = 12.

Q: Can the probability be written as 3/13?
A: Yes, 12/52 simplifies to 3/13. Both forms represent the same probability.

Q: Are Aces included?
A: No. Aces are not classified as face cards, even though many players treat them as special.

Study Tip

When solving card probability problems, start by listing all relevant ranks or suits. Classifying the deck before computing the probability helps prevent mistakes and builds a solid foundation for more advanced topics such as conditional probability and tree diagrams.