GCSE Maths Practice: probability-basics

Question 6 of 11
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\( \begin{array}{l}\textbf{Find the probability of drawing two red cards}\\ \textbf{from a 52-card deck without replacement.}\end{array} \)

Choose one option:

Understanding Probability Without Replacement

In GCSE Maths, questions involving probability "without replacement" are designed to test your ability to update probabilities after each event. When you remove an item from a set, the total number of outcomes changes, so the probability of the next event must be recalculated. This distinguishes Higher-tier questions from simpler Foundation ones, where probabilities usually remain constant.

Key Idea

When choosing cards from a deck without replacement, each draw affects the next. Removing a card changes both the number of favourable outcomes and the total number of possible outcomes.

Step-by-Step Method

  1. Identify the number of favourable outcomes for the first event.
  2. Work out the probability of the first event.
  3. Remove a successful card from the deck to update totals.
  4. Calculate the probability of the second event using the new totals.
  5. Multiply the probabilities because both events must happen in sequence.

Worked Example 1

Suppose you want the probability of drawing two queens without replacement. There are 4 queens in a deck of 52 cards. The probability for the first queen is \(\frac{4}{52}\). After removing one queen, 3 remain out of 51 cards. Multiply: \(\frac{4}{52} \times \frac{3}{51}\).

Worked Example 2

If you want the probability of drawing two black cards, note there are 26 black cards. First draw: \(\frac{26}{52}\). Second draw after removing one: \(\frac{25}{51}\). The approach mirrors the red-card scenario but highlights how the method generalises.

Common Mistakes

  • Using 52 for both probabilities instead of reducing to 51 after the first draw.
  • Forgetting that the number of favourable cards also decreases by one after a successful first event.
  • Adding probabilities when the events happen in sequence — multiplication is required.
  • Assuming the events are independent; in "without replacement" problems, they are not.

Real-Life Applications

This type of probability appears in quality control (drawing samples from a batch), card games, raffle draws, and other situations where something is selected and not returned. Understanding how probabilities adjust builds strong reasoning skills.

FAQ

Q: Why do we multiply the probabilities?
Because both events occur in sequence. This is the rule for dependent events.

Q: What if the first card is not red?
Then the probability for the second event changes. Update after each draw.

Q: Can this method extend to three or more draws?
Yes. Update the totals and multiply each probability.

Study Tip

Visualise the deck shrinking after each draw — this reduces mistakes in dependent probability questions.

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