GCSE Maths Practice: conditional-probability

Question 3 of 10

This question tests conditional probability where more than one outcome satisfies the condition on the second draw.

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\( \begin{array}{l}\text{A bag contains 3 red, 5 green, and 2 blue marbles.} \\ \text{One marble is drawn and not replaced.} \\ \text{What is the probability of drawing a red or blue marble on the second draw,} \\ \text{given that the first draw was green?}\end{array} \)

Choose one option:

Always update the total number of items and list all favourable outcomes before forming the probability.

Conditional Probability with Combined Outcomes

This question focuses on conditional probability where the second event depends on information from the first event. The key phrase given that signals that the situation must be updated before calculating the probability of the next outcome.

Because the marbles are drawn without replacement, the first draw permanently changes the contents of the bag. This means the sample space for the second draw is smaller than the original one, and all probabilities must be calculated using the updated totals.

Understanding the Logical Structure

When solving conditional probability problems, it is helpful to think in terms of a new scenario rather than trying to adjust a formula. Once the first outcome is known, imagine starting a brand-new question using only the remaining objects.

In this question, the second draw is not asking for a single colour but for a combined outcome: red or blue. This requires identifying multiple favourable outcomes and adding them together correctly.

Step-by-Step Strategy

  1. Identify the total number of items originally present.
  2. Use the given condition to remove the known outcome.
  3. Recount the total number of remaining items.
  4. Identify all outcomes that satisfy the condition on the second draw.
  5. Form the probability using remaining favourable outcomes divided by remaining total outcomes.

Worked Example (Different Context)

A jar contains 4 black, 6 white, and 2 yellow beads. One white bead is removed. What is the probability that the next bead chosen is black or yellow?

After removing one white bead, there are 11 beads left. The favourable outcomes are the black and yellow beads, which total 6. The probability is therefore \(\frac{6}{11}\).

Another Example

A pack contains 12 cards numbered 1 to 12. One even number is removed. What is the probability that the next card drawn is a multiple of 3?

The total number of cards is now 11. The favourable outcomes depend on which numbers remain, showing why conditional probability requires careful reasoning rather than shortcuts.

Common Errors

  • Using the original total instead of the updated total.
  • Forgetting that multiple outcomes can be favourable.
  • Subtracting from the wrong colour group.
  • Ignoring the conditional information entirely.

Why This Is Higher Tier

Higher-tier GCSE questions often test reasoning rather than routine calculation. This problem requires interpreting conditions, updating the sample space, and combining probabilities logically — all core Higher skills.

Real-Life Relevance

Conditional probability is used in many real-world situations such as quality inspections, medical diagnosis, and data filtering. Decisions are often made based on updated information rather than initial assumptions.

Study Tip

When a question says given that, pause and rewrite the situation using only what remains. Treat it as a new problem with clearer numbers.

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