Conditional Probability Quizzes

Introduction

Conditional probability is a key concept in GCSE Maths that allows students to calculate the probability of an event occurring given that another event has already happened. Understanding conditional probability is essential for solving real-world problems, interpreting probability tables, tree diagrams, and Venn diagrams, and accurately analysing dependent events.

Core Concepts

What is Conditional Probability?

Conditional probability is the probability of an event A occurring given that another event B has already occurred. It is denoted as:

$$ P(A|B) $$

Where:

• \(P(A|B)\) = Probability of A given B
• A = Event of interest
• B = Event that is known to have occurred

Conditional probability is important when events are dependent, meaning the occurrence of one event affects the probability of the other.

Key Terms

• Independent Events: Events where one does not affect the probability of the other. For independent events, P(A|B) = P(A).
• Dependent Events: Events where one affects the probability of the other. For dependent events, P(A|B) ≠ P(A).
• Sample Space: The set of all possible outcomes.
• Intersection: \(A \cap B\) – outcomes that belong to both events.

Rules & Formula

The formula for conditional probability is:

$$ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{provided } P(B) \neq 0 $$>

Where:

• \(P(A \cap B)\) = Probability that both A and B occur
• \(P(B)\) = Probability that B occurs

Worked Examples

Example 1: Drawing a Card

Experiment: Draw a card from a standard 52-card deck.

Event A: Drawing a king

Event B: Drawing a face card (J, Q, K) → 12 cards

Find P(A|B): probability of drawing a king given the card is a face card.

• P(A ∩ B) = P(king and face card) = 4/52
• P(B) = 12/52

$$ P(A|B) = \frac{4/52}{12/52} = \frac{4}{12} = \frac{1}{3} \approx 0.333 $$

Example 2: Using a Two-Way Table

Survey of 50 students:

	Maths	Science	Total
Male	8	7	15
Female	6	9	15
Total	14	16	30

Find P(student is male | student studies Science):

• P(Male ∩ Science) = 7/30
• P(Science) = 16/30

$$ P(\text{Male | Science}) = \frac{7/30}{16/30} = \frac{7}{16} \approx 0.4375 $$

Example 3: Using a Tree Diagram (Dependent Events)

Experiment: Draw two counters from a bag with 3 red and 2 blue counters without replacement.

Step 1: Tree diagram shows first draw (red or blue) and second draw (adjusted probabilities for dependent event).

Step 2: Event A = second counter is blue, Event B = first counter is red → P(A|B)

• P(A ∩ B) = P(first red and second blue) = (3/5) × (2/4) = 6/20 = 0.3
• P(B) = P(first red) = 3/5 = 0.6

$$ P(\text{second blue | first red}) = \frac{0.3}{0.6} = 0.5 $$

Example 4: Real-Life Scenario

A factory produces 60% red widgets and 40% blue widgets. 10% of red widgets are defective; 5% of blue widgets are defective.

Event A = widget is defective, Event B = widget is red.

$$ P(A|B) = \frac{P(\text{defective and red})}{P(\text{red})} = \frac{0.6 \times 0.1}{0.6} = 0.1 $$>

The probability that a red widget is defective is 10%.

Common Mistakes

• Confusing P(A|B) with P(B|A).
• Failing to identify dependent vs independent events.
• Using total probability of the sample space instead of P(B) in the denominator.
• Misinterpreting “given” in word problems.
• Ignoring adjustments for “without replacement” situations.

Applications

Conditional probability is used widely in exams and real-world contexts:

• Medical testing: Probability of having a disease given a positive test result.
• Quality control: Probability a second item is defective given the first is defective.
• Genetics: Probability of inheriting a trait given parents’ genotypes.
• Surveys and marketing: Probability a customer buys a product given demographic characteristics.
• Weather forecasting: Probability of rain given certain atmospheric conditions.

Strategies & Tips

• Identify clearly what event is “given” (the condition).
• Use Venn diagrams, two-way tables, or tree diagrams to organise information visually.
• Remember the formula: P(A|B) = P(A ∩ B) / P(B)
• Check if events are independent. For independent events, P(A|B) = P(A).
• Practice converting word problems into formulae systematically.
• For dependent events, carefully adjust probabilities after each step.

Summary & Encouragement

Conditional probability allows us to calculate the likelihood of an event given another event has occurred. Key points:

• P(A|B) = P(A ∩ B) / P(B)
• Ensure you correctly identify the “given” event.
• Use Venn diagrams, two-way tables, and tree diagrams to organise information.
• Check whether events are independent or dependent.
• Practice word problems, surveys, and sequential experiments to build confidence.

Work through examples and use diagrams to visualise relationships between events. This will improve your accuracy in GCSE Maths probability questions. Complete the quizzes to reinforce these skills!