GCSE Maths Practice: listing-outcomes

Question 10 of 10

Understand valid outcomes when drawing two cards without replacement from one deck.

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\( \begin{array}{l}\textbf{Which outcomes are valid when drawing two} \\ \textbf{cards from a single deck without replacement?}\end{array} \)

Select all correct options:

Remember: the same card cannot appear twice when no replacement occurs.

Understanding Outcomes When Drawing Two Cards Without Replacement

This question focuses on determining which pairs of cards can occur when drawing two cards from a single standard 52-card deck without replacement. In GCSE Higher probability, it is essential to distinguish between situations where cards are drawn from one deck versus multiple decks, and whether replacement occurs. These details affect whether duplicate cards can appear, how many unique outcomes exist, and what combinations are possible.

A standard deck contains 52 unique cards. Each card appears exactly once: one Ace of Spades, one King of Hearts, one Queen of Diamonds, and so on. When drawing two cards without replacement, the first card is removed from the deck permanently before drawing the second. This means no individual card can be drawn again during the second draw. Therefore, any pair that repeats the exact same card twice is automatically impossible.

Valid Outcomes

The following outcomes are valid:

  • (Ace of Spades, King of Hearts) — both cards exist, both are unique, and the order is allowed.
  • (Queen of Diamonds, King of Clubs) — again, valid and unique cards drawn in sequence.

These represent typical possible outcomes when drawing two cards from a full deck.

Invalid Outcome

  • (Ace of Spades, Ace of Spades) — This is impossible. The deck has only one Ace of Spades, so drawing it twice without replacement cannot occur. This is a common type of error in probability problems, and exam questions often test whether students recognise when duplicate outcomes violate replacement rules.

Step-by-Step Reasoning

  1. Identify each card in the pair.
  2. Check whether both cards exist in the deck.
  3. Check whether the same card appears twice.
  4. If the same card appears twice and there is no replacement allowed, the result is invalid.
  5. All other distinct combinations are valid outcomes.

Worked Example 1: Drawing a Heart Followed by Any King

There are 13 Hearts and 4 Kings in the deck. For any valid pair, the Heart card must be different from the King card. All 13 × 4 = 52 possible ordered pairs are valid because replacement is not required for these to occur — they are different cards.

Worked Example 2: Drawing Two Queens

This is possible: (Queen of Hearts, Queen of Spades), (Queen of Clubs, Queen of Diamonds), etc. There are 4 Queens, so favourable outcomes = 4 × 3 = 12. What is not possible is drawing the same Queen twice, such as (Queen of Hearts, Queen of Hearts).

Worked Example 3: Drawing a Red Card Then Drawing the Same Red Card Again

This is impossible without replacement, because once the first red card is removed, it cannot appear again. However, drawing two different red cards is perfectly valid.

Common Misunderstandings

  • Thinking duplicates are allowed when drawing from one deck. They are not—unless replacement is explicitly mentioned.
  • Confusing rank duplication with full-card duplication. You can draw two Kings, but never the same King twice.
  • Ignoring the importance of order. (Ace of Spades, King of Hearts) is a different outcome from (King of Hearts, Ace of Spades).
  • Mixing up one-deck and two-deck scenarios. Two-deck questions allow duplicates; single-deck questions do not.

Real-Life Applications

This type of reasoning applies to card games, sampling without replacement, probability trees, and dependent-event calculations. When the deck shrinks after each draw, probabilities change. Recognising valid outcomes is the foundation for conditional probability, expected values, and more advanced Chapter 5 GCSE topics.

FAQ

Q: Why is (Ace of Spades, Ace of Spades) invalid?
A: Because there is only one Ace of Spades in the deck.

Q: Would it be valid with replacement?
A: Yes. You could draw the Ace of Spades twice if the first draw were returned to the deck.

Q: Are all pairs of different cards valid?
A: Yes. Any two distinct cards can be drawn from a single deck.

Study Tip

Whenever a question involves drawing cards without replacement, underline the phrase “without replacement”. This is your signal that duplicate card outcomes are impossible. Creating this habit prevents many common mistakes in GCSE exams.

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