E4a — AQA GCSE Statistics
Test yourself on E4a with AQA GCSE practice questions.
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Your focus
- Know and apply the formal notation for independent events.
E4a exam tips
Quick Revision Summary (Key Takeaway)
AQA GCSE Statistics topic E4a covers the addition laws of probability, including mutually exclusive events where P(A or B) = P(A) + P(B) and general non-mutually exclusive events where P(A U B) = P(A) + P(B) - P(A n B). Mastering these rules allows students to accurately calculate union probabilities from Venn diagrams, contingency tables, and contextual problem-solving scenarios.
Topic Overview
Topic E4a focuses on the fundamental addition laws of probability and the structural classification of events. Students learn how to identify mutually exclusive events and apply both the simple addition rule and the general addition rule across multiple representations including tables, lists, and Venn diagrams.
This topic bridges basic single-event probability with advanced bivariate analysis and risk modelling in AQA GCSE Statistics. Understanding when and how to subtract overlapping outcomes is essential for accurate calculations in both theoretical exam scenarios and real-world data interpretation.
Key Concepts
- →Mutually Exclusive Events: Two or more events that cannot occur simultaneously, such that P(A n B) = 0.
- →Addition Law for Mutually Exclusive Events: P(A U B) = P(A) + P(B).
- →General Addition Law: P(A U B) = P(A) + P(B) - P(A n B), used when events can occur together to prevent double-counting.
- →Exhaustive Events: A set of events whose union encompasses the entire sample space, meaning at least one must occur and their total probability sums to 1.
Examiner Tips
- 💡Always state the general formula P(A U B) = P(A) + P(B) - P(A n B) before substituting values, as examiners award method marks even if an arithmetic error occurs later.
- 💡When using two-way contingency tables, identify whether the question asks for an 'or' probability across rows/columns or a conditional probability, and check that totals are used correctly.
- 💡Keep answers as exact fractions in their simplest form unless the question specifies decimals or percentages.
Common Mistakes
- Thinking that 'mutually exclusive' means the same thing as 'independent'. In reality, mutually exclusive events are heavily dependent because if one occurs, the probability of the other occurring becomes 0.
- Forgetting to subtract P(A n B) when calculating 'P(A or B)' from word problems, resulting in double-counting individuals who belong to both categories.
- Assuming that if two events are mutually exclusive, they must also be exhaustive. Two events can be mutually exclusive without covering all possible outcomes.
Revision Plan
- 1Day 1-2: Master definitions of mutually exclusive versus independent events and practice writing formal explanations using set notation.
- 2Day 3-4: Solve pure calculation problems using the general addition law P(A U B) = P(A) + P(B) - P(A n B) with cards, dice, and spinners.
- 3Day 5-6: Practise extracting union and intersection probabilities from Venn diagrams and two-way tables.
- 4Day 7: Complete past paper exam questions under timed conditions, specifically focusing on multi-step context questions.
Exam Question Types
- 📋Categorisation and justification questions asking whether two named events are mutually exclusive with formal mathematical proof.
- 📋Numerical calculation questions requiring the application of the general addition formula using given values or Venn diagrams.
- 📋Contingency table calculations where students must calculate the probability of a compound 'either/or' event.
Command Word Expectations (AQA)
Give a direct decision (e.g. 'Yes' or 'No') accompanied by an explicit statistical reason referencing P(A n B) = 0 or the impossibility of both outcomes coinciding.
Show full working: write down the relevant formula, substitute numerical values, and provide the final answer as a simplified fraction, decimal, or percentage.
Provide a complete algebraic or numerical chain of reasoning leading unambiguously to the given final value without skipping intermediate arithmetic.
How Students Lose Marks (Examiner Pitfalls)
Step-by-Step Worked Solutions
Question: In a group of 80 college students, 45 study Mathematics, 30 study Economics, and 15 study both Mathematics and Economics. A student is selected at random. Calculate the probability that the student studies Mathematics or Economics.
- 1.Step 1: Define events and identify given values. Let M be the event a student studies Mathematics and E be the event a student studies Economics. P(M) = 45/80, P(E) = 30/80, and P(M n E) = 15/80.
- 2.Step 2: Recognize that M and E are not mutually exclusive since 15 students study both. Apply the general addition law: P(M U E) = P(M) + P(E) - P(M n E).
- 3.Step 3: Substitute the known probabilities: P(M U E) = 45/80 + 30/80 - 15/80 = 60/80.
- 4.Step 4: Simplify the fraction to simplest form: 60/80 = 3/4 (or 0.75).
Question: A fair six-sided die is rolled. Event A is rolling an odd number, and Event B is rolling a 6. State, with a mathematical reason, whether events A and B are mutually exclusive. Hence, calculate P(A or B).
- 1.Step 1: List the outcomes for each event. Die sample space S = {1, 2, 3, 4, 5, 6}. Event A = {1, 3, 5} and Event B = {6}.
- 2.Step 2: Check for common outcomes. A n B = {}, so P(A n B) = 0. Because they share no common outcomes and cannot happen together, they are mutually exclusive.
- 3.Step 3: Apply the mutually exclusive addition rule: P(A or B) = P(A) + P(B).
- 4.Step 4: Calculate the individual probabilities and sum them: P(A) = 3/6, P(B) = 1/6. P(A or B) = 3/6 + 1/6 = 4/6 = 2/3.