GCSE Maths Practice: conditional-probability

Conditional Probability with Face Cards

This question uses a familiar playing-card context to illustrate conditional probability. The condition tells us that the card drawn is a face card, which immediately restricts the sample space. Instead of considering all 52 cards in the deck, we now work only with the cards that satisfy this condition.

Face cards are a specific subset of a deck. They consist of kings, queens, and jacks only. Understanding this definition is essential before any probability calculation can be made.

How the Condition Changes the Sample Space

Originally, there are 52 possible outcomes when drawing a card. However, once we are told the card is a face card, all numbered cards and aces are excluded. The new sample space contains only the face cards.

This reduction of the sample space is what makes the probability conditional. All calculations must now be based on the smaller group defined by the condition.

Step-by-Step Method

1. Identify the condition given in the question.
2. Recall or determine which outcomes satisfy the condition.
3. Count how many outcomes are in this restricted group.
4. Count how many of these outcomes meet the required event.
5. Form the probability using the restricted total.

Worked Example 1

A card is drawn from a standard deck and is known to be a red card. What is the probability that it is a heart?

Answer: Once restricted to red cards, the sample space includes hearts and diamonds only. The probability is found by comparing the number of hearts to the total number of red cards.

Worked Example 2

A card is drawn and is known to be a picture card. What is the probability that it is a queen?

Answer: Picture cards are kings, queens, and jacks. The probability is found by counting queens within this group and dividing by the total number of picture cards.

Common Mistakes

• Using 52 as the total instead of the number of face cards.
• Forgetting which cards are classified as face cards.
• Including aces or numbered cards by mistake.
• Confusing unconditional probability with conditional probability.

Real-Life Application

This type of reasoning is used whenever information limits the group you are analysing. For example, if a survey result is known to come from a particular age group, probabilities about preferences should be calculated only within that group. The card example mirrors this idea in a simple and familiar way.

Frequently Asked Questions

Why don’t we use all 52 cards?
Because the condition tells us the card is a face card, all other cards are excluded.

Is this different from normal probability?
Yes. Conditional probability always works with a reduced sample space.

Do I need a formula?
No. At Foundation level, clear reasoning and counting are sufficient.

Study Tip

Whenever a condition is given, pause and rewrite the problem using only the outcomes that are still possible. This makes the correct probability much easier to identify.