M: Probability — AQA A-Level Mathematics
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M: Probability explained
Two events are mutually exclusive if they cannot happen at the same time, so P(A and B) = 0 and P(A or B) = P(A) + P(B).
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Two events are independent if the occurrence of one does not affect the probability of the other, so P(A and B) = P(A) × P(B). You must distinguish these ideas and use them to calculate probabilities. For example, rolling a 3 and rolling a 5 on a single die are mutually exclusive, while rolling a 3 and then a 5 on two dice are independent. These rules extend to discrete distributions such as the binomial, where trials are independent, and to continuous distributions such as the normal, where probabilities are found from areas under the curve.
Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables. Understand and use the conditional probability formula. P A B = P A ∩B P B
Conditional probability is the probability of an event given that another event has already occurred. You must understand and use it with tree diagrams, Venn diagrams and two-way tables, and apply the formula P(A|B) = P(A ∩ B) / P(B). For example, if a bag has 3 red and 2 blue counters, the probability of a second red given a first red is 2/4. Tree diagrams show this by changing probabilities on the second set of branches. Venn diagrams show it as the proportion of B that lies inside A. Two-way tables give counts, so you divide the count in the overlap by the total for the condition. The formula works when P(B) > 0. Always identify the condition and restrict the sample space to that event.
Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.
Modelling with probability means using probability models to represent real-world situations. You must critique the assumptions made in a model and discuss how more realistic assumptions would affect the model's predictions. For example, assuming a coin is fair gives a probability of 0.5 for heads, but if the coin is biased, the probability changes. When critiquing, identify each assumption, explain why it may not hold, and state whether the true probability would be higher or lower. For instance, assuming independent trials in a medical test may be unrealistic if the disease spreads, leading to clustered results. More realistic assumptions often make the model more complex but more accurate. You should be able to suggest improvements and describe the likely direction of change in the calculated probabilities.
Your focus
- Define mutually exclusive and independent events and identify them in given contexts.
- Calculate probabilities using the addition rule for mutually exclusive events and the multiplication rule for independent events.
- Explain the difference between mutually exclusive and independent events, including why mutually exclusive events with non-zero probabilities are not independent.
Show all 10 objectives
- Apply probability rules to problems involving discrete distributions such as the binomial and continuous distributions such as the normal.
- Calculate conditional probabilities using the formula P(A|B) = P(A ∩ B) / P(B).
- Interpret and construct tree diagrams, Venn diagrams and two-way tables to solve conditional probability problems.
- Distinguish between P(A|B) and P(B|A) and apply the correct one in context.
- Identify and critique the assumptions made in a probability model.
- Explain how changing assumptions to be more realistic would affect the probabilities calculated.
- Suggest improvements to a probability model and discuss the implications of those changes.
M: Probability exam tips
Marking Points
- State that mutually exclusive events have no intersection and use P(A or B) = P(A) + P(B) for such events.
- State that independent events satisfy P(A and B) = P(A) × P(B) and use this to calculate combined probabilities.
- Distinguish clearly between mutually exclusive and independent events, explaining that mutually exclusive events with non-zero probabilities cannot be independent.
- Apply the addition and multiplication rules to problems involving two or more events, including those expressed in words or tables.
- Link independence to discrete distributions, for example recognising that binomial trials are independent and using this to justify the model.
- Link probability calculations to continuous distributions, for example using the normal distribution to find probabilities and interpreting them in context.
- Correctly identify the condition event and restrict the sample space to that event when calculating a conditional probability.
- Use the formula P(A|B) = P(A ∩ B) / P(B) accurately, ensuring that P(B) is not zero.
- Interpret and construct tree diagrams where probabilities on the second set of branches are conditional on the first outcome.
- Use Venn diagrams to find conditional probabilities by dividing the probability of the intersection by the probability of the condition.
- Extract conditional probabilities from two-way tables by dividing the relevant cell count by the row or column total for the condition.
- Recognise and distinguish between P(A|B) and P(B|A), and between conditional and unconditional probabilities.
- Identify the assumptions made in a given probability model, such as independence, equal likelihood, or constant probability.
- Critique each assumption by explaining why it might not be valid in the real-world context.
- Describe the likely effect of relaxing an assumption on the calculated probabilities, stating whether they would increase or decrease.
- Suggest more realistic assumptions and explain how they would alter the model or its predictions.
- Evaluate the trade-off between simplicity and accuracy when choosing a probability model.
- Communicate critiques clearly, using appropriate probabilistic language and referring to the context.
Examiner Tips
- 💡Write down the rule you are using before substituting numbers, so the examiner can follow your reasoning.
- 💡For 'or' questions, check whether the events are mutually exclusive; if not, subtract the overlap P(A and B).
- 💡For 'and' questions, check whether the events are independent; if not, use conditional probability.
- 💡When linking to distributions, name the distribution and its parameters, and interpret the resulting probability in the context of the question.
- 💡Always write down the formula P(A|B) = P(A ∩ B) / P(B) before substituting values; this makes your method clear and can earn method marks.
- 💡When using a tree diagram, label each branch with its probability and remember that second-level probabilities are conditional on the first outcome.
- 💡For two-way tables, annotate the table with row and column totals to help you pick the correct denominator for the condition.
- 💡Check whether events are independent: if P(A|B) = P(A), then A and B are independent, which can simplify calculations.
- 💡Read the context carefully and list all assumptions before writing your critique; this ensures you address the modelling aspect fully.
- 💡Use phrases like 'assuming independence' or 'assuming a fair coin' to make your assumptions explicit.
- 💡When suggesting a more realistic assumption, always state the likely effect on the probability, using comparative language such as 'higher' or 'lower'.
- 💡Link your critique back to the original problem to show you understand the practical implications.
Common Mistakes
- Confusing mutually exclusive with independent. Correction: mutually exclusive means they cannot both occur; independent means one does not change the other's probability.
- Adding probabilities when events are independent and you need both to occur. Correction: use multiplication for 'and' with independent events, and addition for 'or' with mutually exclusive events.
- Assuming events are independent without justification. Correction: check whether the context implies independence, or use the condition P(A and B) = P(A) × P(B) to test it.
- Applying discrete rules directly to continuous distributions without using the appropriate distribution model. Correction: for continuous distributions, use the given distribution or normal probabilities, not simple counting.
- Confusing P(A|B) with P(B|A). Correction: always check which event is given; the condition is the event that has already happened.
- Using the wrong denominator in the conditional probability formula, such as dividing by P(A) instead of P(B). Correction: the denominator is always the probability of the condition.
- Forgetting to update probabilities on the second set of branches of a tree diagram when sampling without replacement. Correction: adjust the numerator and denominator to reflect the reduced sample space.
- Misreading a two-way table by using the grand total instead of the row or column total for the condition. Correction: identify the condition and use the corresponding marginal total.
- Failing to identify any assumptions and instead just describing the model. Correction: explicitly list assumptions such as independence or fairness before critiquing.
- Critiquing an assumption without stating the effect on the probability. Correction: always say whether the probability would be higher or lower and why.
- Assuming that more realistic models are always better without considering practicality. Correction: discuss both accuracy and simplicity, and note that added complexity may not be justified.
- Using vague language such as 'it might change' without direction. Correction: be specific, e.g. 'the probability would increase because...'.