GCSE Maths Practice: tree-diagrams

Question 11 of 11

Two cards are drawn from a standard deck without replacement.

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\( \begin{array}{l}\textbf{Two cards are drawn from a standard deck of 52 cards without replacement.}\\ \text{Which is more likely: drawing a red card then a black card, or drawing two black cards?}\end{array} \)

Choose one option:

Calculate both probabilities fully and compare them using a common denominator.

Comparing Probabilities Using Tree Diagrams

This question goes beyond basic probability calculations and is suitable for GCSE Higher tier because it requires comparison and reasoning, not just finding a single probability. You are asked to decide which of two different events is more likely.

Understanding the Situation

A standard deck of cards contains 52 cards: 26 red (hearts and diamonds) and 26 black (clubs and spades). Two cards are drawn one after another without replacement, meaning the first card is not put back before the second card is drawn. This makes the events dependent, because the probabilities change after the first draw.

Breaking the Problem into Two Cases

To answer the question, you must calculate and compare two different probabilities:

  • The probability of drawing a red card first and then a black card
  • The probability of drawing two black cards

Each case must be worked out carefully, taking into account how the deck changes after the first card is drawn.

Why This Is a Higher-Tier Question

At Foundation level, students are usually asked to calculate a single probability. Here, you must calculate two probabilities and then compare them. This requires a deeper understanding of conditional probability and careful handling of fractions.

Using Tree Diagrams

Tree diagrams are a helpful way to organise this information. One tree can show the outcomes for drawing a red card first, followed by a black card. Another tree (or branch) can show the outcomes for drawing two black cards. Multiplying along each path gives the probability for that sequence.

Worked Example (Different Numbers)

A bag contains 5 blue balls and 7 green balls. Two balls are taken without replacement. Compare the probability of drawing a blue then a green with the probability of drawing two green balls.

  • P(Blue then Green) = 5/12 × 7/11
  • P(Green then Green) = 7/12 × 6/11

To decide which is larger, the fractions must be compared carefully.

Common Mistakes

  • Assuming the two probabilities are equal because the deck starts with equal numbers of red and black cards
  • Forgetting to reduce the number of available cards after the first draw
  • Comparing fractions without converting them to a common denominator

Exam Tip

When asked which event is more likely, always calculate both probabilities fully and compare them using equivalent fractions or a common denominator. A clear comparison is essential for full marks.

Study tip: If a question asks you to compare probabilities, it is almost always Higher tier and requires more than one calculation.

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