GCSE Maths Practice: listing-outcomes

Understanding Independent Events with Two Separate Decks

This question focuses on identifying valid outcomes when drawing one card from each of two separate 52-card decks. Because the decks are independent and identical, the possibilities greatly expand compared to drawing two cards from the same deck. In probability, independence means the outcome of one event does not affect the outcome of the other. When two identical decks are used, every card in Deck A has a matching copy in Deck B, and each card exists once per deck. Therefore, combinations that would normally be impossible in a single-deck scenario become fully valid when using two decks.

For example, drawing the Ace of Hearts from Deck A and the Ace of Hearts from Deck B is entirely possible, because each deck contains its own Ace of Hearts. This differs from a single-deck situation, where drawing the same card twice without replacement is impossible. Higher-tier GCSE questions often test whether students can identify when duplication is or is not allowed, depending on whether replacement or independence is specified.

Step-by-Step Reasoning

1. Each deck contains 52 distinct cards.
2. You draw one card from each deck.
3. The decks do not interact with each other.
4. Because they are independent, any card from Deck A can pair with any card from Deck B.
5. Therefore, all combinations of two cards—including identical ones—are valid outcomes.

Why All Provided Outcomes Are Valid

Let’s examine the outcomes listed:

• (Ace of Hearts, 2 of Spades): A normal combination of two different cards across two decks.
• (King of Diamonds, King of Clubs): Both are face cards from Deck A and Deck B, each a valid selection.
• (Ace of Hearts, Ace of Hearts): Only possible because the decks are separate and each contains its own Ace of Hearts.

All three combinations can genuinely occur when drawing independently from two decks.

Worked Example 1: Drawing Two Cards Without Replacement From One Deck

In a single-deck situation, drawing (Ace of Hearts, Ace of Hearts) without replacement would be impossible. This contrast reinforces the importance of paying attention to the number of decks.

Worked Example 2: Probability of Both Cards Being Queens

Each deck has 4 Queens. So favourable outcomes = 4 × 4 = 16. Total outcomes = 52 × 52 = 2704. Probability = 16/2704 = 1/169.

Worked Example 3: Probability of Drawing a Heart from Deck A and a Red Card from Deck B

Deck A has 13 Hearts; Deck B has 26 red cards. So probability = (13/52) × (26/52) = 1/4.

Common Misunderstandings

• Thinking duplicated cards are impossible. They are impossible only when cards are drawn from the same deck without replacement.
• Confusing independence with replacement. Independence comes from using two separate decks, not from placing cards back.
• Assuming outcomes must be unique across decks. This is not true—identical outcomes are expected in multi-deck probability.
• Believing the order of decks doesn’t matter. The first card comes from Deck A, the second from Deck B. The sequence is ordered.

Real-Life Applications

Independent deck scenarios appear in card games, simulations, random sampling, and Monte-Carlo modelling. Independence is a foundational concept in statistics, where events must be analysed correctly based on whether one outcome influences another. Understanding how independence works in physical examples like card decks strengthens a student’s ability to interpret real data and complex probability structures.

FAQ

Q: Why can the same card be drawn twice?
A: Each deck has its own copy of every card.

Q: Are the decks considered identical?
A: Yes, standard decks have the same 52 cards.

Q: Is the order important?
A: Yes. (Ace of Hearts, 2 of Spades) is different from (2 of Spades, Ace of Hearts).

Study Tip

When a probability question mentions “two decks”, immediately recognise that duplicated cards are possible. This small detail changes the entire structure of the outcome space and prevents incorrect assumptions.