GCSE Maths Practice: listing-outcomes

Question 7 of 10

Practise working with overlapping sets to find probabilities involving unions of events.

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\( \begin{array}{l}\textbf{What is the probability of drawing a red card} \\ \textbf{or a face card from a 52-card deck?}\end{array} \)

Choose one option:

Always subtract overlapping outcomes to avoid counting them twice.

Understanding Probability with Combined Events in a Deck of Cards

This question explores a key Higher GCSE concept: calculating the probability of a combined event using the inclusion–exclusion principle. When an event involves the word “or”, you are dealing with the union of two sets. In probability, the formula is:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

This prevents double-counting outcomes that appear in both sets. Here, the two relevant sets are the set of all red cards and the set of all face cards. Because some cards belong to both groups (the red face cards), the overlap must be subtracted.

Step-by-Step Counting

  1. The deck contains 52 cards in total.
  2. There are 26 red cards (13 Hearts + 13 Diamonds).
  3. There are 12 face cards (Jack, Queen, King in each suit).
  4. Of these face cards, 6 are red (♥J, ♥Q, ♥K, ♦J, ♦Q, ♦K).
  5. Use inclusion–exclusion: 26 + 12 − 6 = 32 favourable outcomes.
  6. Write the probability as 32/52.

This reasoning is essential for higher-tier probability, where overlapping sets frequently appear.

Worked Example 1: Probability of Drawing a Black Card or a Queen

There are 26 black cards (Spades and Clubs) and 4 Queens altogether. Among these, 2 Queens are black. Applying inclusion–exclusion: 26 + 4 − 2 = 28 favourable outcomes. Probability = 28/52.

Worked Example 2: Probability of Drawing a Diamond or a Face Card

Diamonds: 13 cards. Face cards: 12 cards. Overlap: 3 (J♦, Q♦, K♦). Total favourable: 13 + 12 − 3 = 22. Probability = 22/52.

Worked Example 3: Probability of Drawing a Red Card or an Ace

Red cards: 26. Aces: 4. Overlap: 2 (Ace of Hearts and Ace of Diamonds). Total favourable: 26 + 4 − 2 = 28. Probability = 28/52.

Common Misunderstandings

  • Forgetting to subtract overlap. Students often add the sets without removing duplicated cards, leading to counts like 38 instead of 32.
  • Incorrectly including Aces with face cards. Aces are not face cards in mathematics.
  • Assuming suits have different numbers of face cards. Each suit has exactly three.
  • Not identifying the meaning of “or”. The word “or” always means a union of sets, not selecting exactly one.

Real-Life Applications

The inclusion–exclusion principle is widely used in computer science (search filters, data overlap), statistics (survey overlaps), and probability-based games. Understanding overlapping sets helps when analysing card games, working with probability trees, solving Venn diagram problems, and calculating combined event likelihoods in real scenarios.

FAQ

Q: Why do red face cards cause double-counting?
A: Because they belong to both groups — red cards and face cards.

Q: Does the fraction simplify?
A: Yes. 32/52 simplifies to 8/13, but unsimplified forms are acceptable unless the exam insists.

Q: Are Jokers included?
A: No. Standard GCSE decks have 52 cards.

Study Tip

Whenever a probability question includes the word “or”, sketch a quick Venn diagram. It helps you visualise the overlap and prevents double-counting.

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