GCSE Maths Practice: conditional-probability

Question 11 of 13

This question demonstrates conditional probability by restricting attention to red cards in a deck.

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\( \begin{array}{l}\text{A standard deck of 52 cards has 4 kings.} \\ \text{What is the probability of drawing a king, given that the card drawn is red?}\end{array} \)

Choose one option:

Always restrict the sample space to match the condition before calculating probability.

Conditional Probability with Playing Cards

Playing cards are commonly used in probability questions because they provide a clear and well-defined sample space. In this question, conditional probability is used to find the likelihood of one property (being a king) given that another property (being red) is already known.

The key phrase is "given that the card drawn is red". This condition immediately restricts the sample space. Instead of considering all 52 cards in the deck, we now work only with the red cards.

Restricting the Sample Space

A standard deck contains 52 cards split evenly between red and black. Once we are told the card is red, all black cards become irrelevant. The new sample space contains only red cards.

Within this restricted group, we then count how many cards satisfy the condition of being a king. Conditional probability always works in two stages: first restrict, then count.

Step-by-Step Method

  1. Identify the condition given in the question.
  2. Limit the sample space to outcomes that satisfy the condition.
  3. Count the total number of outcomes in this restricted group.
  4. Count how many of those outcomes meet the required event.
  5. Write the probability using the restricted total.

Worked Example 1

A standard deck of cards is used. What is the probability of drawing a queen, given that the card is black?

Answer: There are 26 black cards. Among them are 2 queens (clubs and spades). The probability is calculated using these values.

Worked Example 2

A card is drawn from a deck. What is the probability it is a heart, given that it is a face card?

Answer: First restrict to face cards only. Then count how many of those are hearts and compare this to the total number of face cards.

Common Mistakes

  • Using 52 as the total instead of the restricted total.
  • Forgetting that the condition changes the sample space.
  • Counting kings of both colours instead of just red kings.
  • Mixing up unconditional and conditional probabilities.

Real-Life Connection

Conditional probability with overlapping characteristics is used in many real-world situations. For example, if a company knows a customer is from a particular region, probabilities about preferences or behaviour are calculated only within that group. The same logic applies in surveys, risk assessment, and data analysis.

Frequently Asked Questions

Why don’t we divide by 52?
Because the condition tells us the card is red, black cards are no longer possible.

Is this different from normal probability?
Yes. Conditional probability always uses a restricted sample space.

Do I need a formula?
No. At Foundation level, careful counting is sufficient.

Study Tip

Always ask yourself: what outcomes are still possible? This helps you identify the correct total for conditional probability questions.

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