Probability Fundamentals

Statistics AQA
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ST19: Probability Fundamentals

Edexcel 1ST0 & AQA 8382

Covers the foundational principles of probability including the 0–1 probability scale, expected frequency, relative risk, absolute risk, experimental versus theoretical probability, and the use of tree diagrams and Venn diagrams to calculate probabilities for combined events.

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Probability Fundamentals

Covers the foundational principles of probability including the 0–1 probability scale, expected frequency, relative risk, absolute risk, experimental versus theoretical probability, and the use of tree diagrams and Venn diagrams to calculate probabilities for combined events.

Key Fact: All probabilities lie on the 0–1 scale: 0 means impossible, 1 means certain, 0.5 means even chance
Key Fact: P(event) = number of favourable outcomes ÷ total number of equally likely outcomes for theoretical probability
Key Fact: Expected frequency = probability × number of trials; e.g. if P(green) = 0.3 and you spin 200 times, expect 60 greens
Key Fact: Relative risk = P(event in group A) ÷ P(event in group B); a relative risk of 2 means group A is twice as likely to experience the event
Key Fact: Absolute risk = P(event occurring); absolute risk difference = P(event in A) − P(event in B) gives the actual change in likelihood
Key Fact: Experimental probability (relative frequency) = frequency of outcome ÷ total trials; it tends towards theoretical probability as trials increase (law of large numbers)
Key Fact: For mutually exclusive events: P(A or B) = P(A) + P(B); they cannot happen simultaneously so there is no overlap
Key Fact: For independent events: P(A and B) = P(A) × P(B); the outcome of one does not affect the other
Key Fact: Tree diagrams show all possible outcomes for two or more stages; multiply along branches for ‘and’ probabilities, add across end branches for ‘or’ probabilities
Key Fact: Venn diagrams use overlapping circles to represent sets; P(A∪B) = P(A) + P(B) − P(A∩B) for non-mutually exclusive events
Key Fact: Conditional probability: P(A|B) = P(A∩B) ÷ P(B); the probability of A given that B has already occurred
Key Fact: Sample space diagrams list all possible outcome pairs in a grid; each cell is equally likely so counting favourable cells gives the probability

📋 Key Vocabulary and Concepts

For Probability Fundamentals, you must know:

❓ Practice Questions

Q: A fair six-sided die is rolled 300 times. What is the expected frequency of rolling a 4?

Q: In a medical trial, 40 out of 1000 people taking a drug experienced side effects, compared with 80 out of 1000 taking a placebo. Calculate the relative risk.

Q: A bag contains 4 red and 6 blue counters. Two counters are drawn without replacement. Draw a tree diagram and find P(both red).

Q: In a Venn diagram, P(A) = 0.6, P(B) = 0.5 and P(A∩B) = 0.2. Find P(A∪B).

Q: A coin is tossed 500 times and heads appears 270 times. Find the experimental probability of heads. Is the coin fair?

✅ Answers

  1. P(4) = 1/6. Expected frequency = (1/6) × 300 = 50
  2. Relative risk = P(side effects with drug) ÷ P(side effects with placebo) = (40/1000) ÷ (80/1000) = 0.04 ÷ 0.08 = 0.5. The drug group has half the risk.
  3. P(R then R) = (4/10) × (3/9) = 12/90 = 2/15 ≈ 0.133
  4. P(A∪B) = P(A) + P(B) − P(A∩B) = 0.6 + 0.5 − 0.2 = 0.9
  5. Experimental probability = 270/500 = 0.54. Expected for fair coin = 0.5. The result is close to 0.5 but slightly high; with 500 trials this small difference could still be due to chance.

🎯 Exam Tips

📝 Exam Technique

GCSE Statistics Exam Tips — Probability Fundamentals:
1. For Probability Fundamentals questions, show every step of your working clearly — method marks count even if the final answer is wrong
2. Check your answer makes sense in context (estimation, units, reasonableness)
3. Use correct mathematical notation and state formulae before substituting values
4. If a Probability Fundamentals question asks you to 'prove' or 'show', write a logical chain of reasoning with a conclusion line
5. For problem-solving, identify the topic first, then recall the relevant method

⚠️ Common Errors

✗ Adding probabilities that are not mutually exclusive without subtracting the overlap ✓ Use P(A∪B) = P(A) + P(B) − P(A∩B) when events can both occur

✗ Multiplying probabilities without checking independence first ✓ Only multiply for ‘and’ when events are independent; for dependent events, adjust the second probability

✗ Confusing relative risk and absolute risk difference ✓ Relative risk is a ratio (division); absolute risk difference is P(A) − P(B) (subtraction)

✗ Forgetting that probabilities on branches from the same node in a tree diagram must sum to 1 ✓ Always check: the probabilities of all branches from one node add to 1

✍️ Model Answer

Full-Mark Response

A game uses a biased coin where P(heads) = 0.6. The coin is tossed twice. Calculate the probability of getting: (a) two heads, (b) at least one tail.

(a) P(H and H) = 0.6 × 0.6 = 0.36 (b) P(at least one tail) = 1 − P(both heads) = 1 − 0.36 = 0.64 Alternatively using a tree diagram: • P(H,T) = 0.6 × 0.4 = 0.24 • P(T,H) = 0.4 × 0.6 = 0.24 • P(T,T) = 0.4 × 0.4 = 0.16 P(at least one tail) = 0.24 + 0.24 + 0.16 = 0.64 Using the complement is quicker and reduces the chance of missing an outcome.

📊 AO Deep Dive

Assessment Objective Analysis

AO1 (Knowledge & Understanding): Demonstrate knowledge and understanding of probability fundamentals, including data collection, presentation and calculation techniques relevant to Edexcel 1ST0 & AQA 8382.

AO2 (Application): Apply knowledge and understanding of probability fundamentals to interpret data, reason statistically and draw conclusions in context.

AO3 (Evaluation): Evaluate statistical methods and conclusions, assessing appropriateness, reliability, validity and bias through the statistical enquiry cycle.

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