GCSE Maths Practice: tree-diagrams

Tree Diagrams with Cards (Higher GCSE)

Card problems are a great way to practise tree diagrams because the probabilities change in a predictable way when you draw without replacement. Each time you take a card from the deck, two things happen: the total number of cards decreases by 1, and the number of favourable cards may or may not change depending on what you drew.

A useful way to think about a tree diagram is: each branch represents a “story” of what happened so far. The probability written on a branch must match that story. For example, if the first draw is an Ace, then the story is “one Ace has been removed”. That means the denominator becomes 51, and the numerator for Aces becomes 3 (because only 3 Aces remain). However, if your next target is Kings, the numerator for Kings might stay at 4 if you didn’t remove any Kings on the first draw. This idea—numerator changes only if you removed one of the favourable outcomes—is a key Higher skill.

With tree diagrams, you usually do two types of calculations:

• Single path questions: multiply along one route (e.g., “Ace then King”).
• Multiple path questions: multiply along several routes and add them (e.g., “Ace and King in any order”, or “exactly one face card”).

Another Higher detail is recognising when events are dependent. Drawing without replacement makes events dependent, because the second probability depends on what happened first. This is why the second branch is written as a conditional probability, such as P(King on the second draw | Ace on the first draw). Even if you don’t write the conditional notation, your fractions on the tree must reflect it.

Worked Example (Different to the Question)

A deck has 4 Queens. Two cards are drawn without replacement. Find the probability of drawing a Queen and then another Queen.

First draw: P(Queen) = 4/52. After removing one Queen, there are 3 Queens left out of 51 cards, so P(Queen | Queen first) = 3/51. Multiply: (4/52)×(3/51).

Common Mistakes

• Forgetting to change the denominator from 52 to 51 on the second draw.
• Changing the numerator when it should stay the same (e.g., drawing an Ace does not change the number of Kings).
• Mixing up “then” with “in any order” (the second requires adding two paths).

If you draw a quick two-stage tree and label each branch clearly, these problems become routine. The tree keeps your logic organised and makes it much easier to avoid slips under exam pressure.