GCSE Maths Practice: probability-basics

Understanding Combined Events in GCSE Probability

This question is an example of a higher-tier GCSE Maths probability problem involving the union of two events. When events can overlap, such as “drawing a diamond” and “drawing a face card,” you cannot simply add the totals together because some cards belong to both categories. To avoid counting these cards twice, you use the addition rule for probability, also known as the inclusion–exclusion principle.

The Two Events

The events here are:

• Drawing a diamond → 13 cards
• Drawing a face card (Jack, Queen, King in any suit) → 12 cards

Face cards exist in all four suits, including diamonds. Therefore, the two events overlap.

Why Overlap Matters

If you were to add 13 diamonds and 12 face cards, you would get 25. However, this total counts the diamond face cards twice (Jack of diamonds, Queen of diamonds, King of diamonds). These three cards belong to both groups, so they must be subtracted once to avoid double counting.

The Inclusion–Exclusion Formula

For two events A and B, the number of outcomes in their union is:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

This rule ensures accuracy whenever events can overlap, and it is essential in higher-tier probability questions involving cards, sets, or Venn diagrams.

Worked Example 1: Drawing a Heart or a Face Card

There are 13 hearts and 12 face cards. The overlap is 3 face hearts. Using the rule:

n(A ∪ B) = 13 + 12 − 3 = 22.

This mirrors the structure of the original problem and reinforces the idea of adjusting for overlap.

Worked Example 2: Drawing a Red Card or a King

There are 26 red cards (hearts and diamonds) and 4 Kings. Two of these Kings are red (King of hearts, King of diamonds). Therefore:

26 + 4 − 2 = 28 favourable outcomes.

Probability = 28/52 = 1/2.

Worked Example 3: Drawing a Spade or a Number Card

Spades = 13 cards. Number cards (2–10) across all suits = 36 cards. The overlap: number spades = 9. Using the rule:

13 + 36 − 9 = 40.

Probability = 40/52.

Common Mistakes

• Double counting: forgetting to subtract overlapping cards.
• Incorrect face-card count: assuming Aces are face cards (they are not).
• Using multiplication instead of addition: multiplication only applies to independent events, not overlapping sets.
• Forgetting card structure: each suit always has Jack, Queen, King, giving exactly 12 face cards total.

Real-Life Applications

This type of reasoning appears in data science, computer algorithms, medical testing, and any field involving classification. The inclusion–exclusion principle helps avoid over-counting when groups overlap. In advanced mathematics and statistics, this rule extends to three or more sets, but GCSE builds the foundation with two-event problems like this one.

FAQ

Q: Why can't I simply add the number of diamonds and face cards?
Because they overlap. The three diamond face cards would be counted twice.

Q: Are Aces face cards?
No. Face cards are Jack, Queen and King only.

Q: Can I simplify 22/52?
Yes. It simplifies to 11/26, but unless the question requires simplest form, the unsimplified fraction is acceptable.

Study Tip

Whenever two events overlap, draw a quick Venn diagram. Label the overlap first, then fill in the remaining values. This simple method prevents counting errors and makes higher-tier probability problems much easier.