Probability

OCR

A-Level

Probability is the mathematical study of uncertainty, quantifying the likelihood of events occurring on a scale from 0 (impossible) to 1 (certain). It encompasses both theoretical probability, based on equally likely outcomes, and experimental probability, derived from relative frequency and large data sets. Mastery of this topic requires the ability to model single and combined events using sample spaces, Venn diagrams, and tree diagrams to calculate probabilities for independent, mutually exclusive, and conditional scenarios.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Subtopics in this area

Probability

Learning Objectives

What you need to know and understand

Use tree diagrams to record probabilities of successive events.
Understand the concept of conditional probability and calculate it from first principles.

Example Examiner Feedback

Real feedback patterns examiners use when marking

"You have correctly calculated the intersection, but check if the events are stated as independent before multiplying probabilities."
"Your Venn diagram is accurate. Now, use the region representing 'B' as your denominator for the conditional probability P(A|B)."
"Excellent use of the addition formula. To secure full marks, ensure you explicitly state the values substituted before the final answer."
"You confused mutually exclusive with independent. Remember: mutually exclusive means P(A ∩ B) = 0; independent means P(A ∩ B) = P(A)P(B)."
"You have correctly identified the intersection, but check your denominator for the conditional probability — which event is 'given'?"
"Your Venn diagram is accurate, but you must explicitly write out the check for independence: does P(A) x P(B) equal the intersection?"
"Be careful with notation; ensure you are distinguishing clearly between union (U) and intersection (n)"
"Excellent use of the addition formula; to secure full marks, ensure you justify why the intersection was subtracted"
"You correctly identified the intersection, but check your denominator — for 'given that', are we looking at the whole group or a subset?"
"Your Venn diagram is accurate. To secure the explanation mark, explicitly state why P(A ∩ B) ≠ 0 means the events are not mutually exclusive."
"Be careful with notation: P(A U B) refers to 'A or B', while P(A n B) refers to 'A and B'."
"You assumed independence here. Unless the question states it, you must calculate P(A ∩ B) from the diagram first."
"You have correctly calculated the individual probabilities, but remember that 'without replacement' changes the total number of items for the second pick."
"Avoid using ratio notation like 1:5 for probability; OCR requires fractions, decimals, or percentages to award the mark."
"Good use of the tree diagram. To access higher marks, ensure you clearly show the multiplication of branches for the combined event."
"You identified the mutually exclusive events correctly; now ensure you add their probabilities rather than multiplying."

Marking Points

Key points examiners look for in your answers

Probabilities must be expressed as fractions, decimals, or percentages; ratios (e.g., 1:4) are generally not accepted for final answers.
Method marks (M) are awarded for showing the correct calculation path, such as multiplying probabilities along tree branches.
In 'Show that' questions, every step of the calculation must be written out explicitly; jumping to the answer loses credit.
For Venn diagrams, marks are awarded for placing elements in the correct regions, specifically handling the intersection correctly.
Award M1 for stating the correct addition formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B) before substitution
Award A1 for correctly calculating conditional probability P(A|B) using the restricted sample space or formula P(A ∩ B) / P(B)
Credit B1 for a fully labeled Venn diagram where all probabilities sum to exactly 1
Award M1 for explicitly testing independence by comparing the product P(A) × P(B) with P(A ∩ B)
Award 1 mark for interpreting the intersection of non-independent events correctly in context
Award M1 for correct statement and substitution into the addition law $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Award A1 for accurate calculation of conditional probability, specifically ensuring the denominator corresponds to the 'given' event
Credit responses that explicitly verify independence by showing $P(A \cap B) = P(A) \times P(B)$ rather than stating it intuitively
Award B1 for a fully labeled Venn diagram where all regions, including the region outside the sets, sum to exactly 1
M1: Award for stating and applying the addition rule P(A ∪ B) = P(A) + P(B) - P(A ∩ B) correctly
M1: Award for correct substitution into the conditional probability formula P(A|B) = P(A ∩ B) / P(B)
A1: Award for accurate calculation of probabilities from a fully labelled Venn diagram or tree diagram
B1: Credit clear justification of independence by showing P(A ∩ B) = P(A) × P(B) explicitly
M1: Award for forming a correct algebraic equation involving unknown probabilities (e.g., x) to satisfy given conditions
Award 1 mark for probabilities expressed correctly as fractions, decimals, or percentages; do not credit ratios (e.g., 1:5) or words (e.g., '1 out of 5').
Award method marks for multiplying probabilities along tree diagram branches for combined events (AND rule).
Credit responses that demonstrate the sum of probabilities for mutually exclusive events equals 1.
For 'without replacement' scenarios, candidates must show the denominator reducing by 1 for the second event to gain method marks.

Examiner Tips

Expert advice for maximising your marks

💡When asked for the probability of 'at least one', calculate 1 minus the probability of 'none' to save time and reduce error.
💡Always check that the sum of probabilities on a set of tree diagram branches equals 1 before moving on.
💡If the question involves 'frequency' or 'experimental data', use the term 'estimate' in your reasoning, as relative frequency is an estimate of probability.
💡Draw a Venn diagram immediately for any question involving 'given that' or overlapping sets to visualize the reduced sample space
💡When asked to 'Show that' events are independent, you must explicitly calculate P(A) × P(B) and compare it numerically to P(A ∩ B) with a concluding statement
💡Pay close attention to the phrase 'given that'; this signals a change in the denominator to the probability of the condition
💡Use set notation (∪, ∩, ') precisely; OCR penalizes ambiguous notation in 'Show that' questions
💡When asked to 'Show that' events are independent, you must calculate the product of individual probabilities and explicitly compare it to the intersection; a statement without calculation gains no credit
💡Draw a Venn diagram immediately for any question involving three categories or phrases like 'at least one' or 'neither' to visualize the sample space
💡Pay close attention to the notation $P(A' \cap B)$ versus $P(A \cap B')$; the position of the prime symbol changes the region entirely
💡When a question includes the phrase 'given that', immediately write down P(A|B) = P(A ∩ B) / P(B) to structure your working and identify the correct denominator.
💡Always draw a Venn diagram for questions involving two or three overlapping events; it is the most reliable method to visualize regions and calculate P(A ∪ B) or P(A' ∩ B).
💡Do not assume events are independent unless stated; you must prove it using P(A ∩ B) = P(A) × P(B) or P(A|B) = P(A).
💡Use set notation (∪, ∩, ') precisely in your working to ensure method marks are awarded even if arithmetic errors occur.
💡If a question asks for an 'estimate' of probability, use the relative frequency from the data provided, not a theoretical assumption.
💡When calculating 'at least one', it is often faster and less prone to error to calculate 1 - P(none).
💡Always check that the branches of your tree diagram sum to 1 at each junction; this is a vital self-check mechanism.
💡In Venn diagram questions, check if the value given is for the intersection (A and B) or the whole set A; subtract the intersection from the total of A to find 'A only'.

Common Mistakes

Pitfalls to avoid in your exam answers

Writing probabilities as ratios (e.g., 3:4) instead of fractions (3/7), which results in zero marks for the final answer.
Failing to reduce the denominator in tree diagrams involving 'without replacement' scenarios.
Confusing the 'AND' rule (multiplication) with the 'OR' rule (addition) in complex multi-stage events.
Incorrectly identifying the region for 'A union B' in Venn diagrams by double-counting the intersection.
Assuming events are independent without verification, leading to the incorrect use of P(A ∩ B) = P(A)P(B)
Confusing 'mutually exclusive' (intersection is zero) with 'independent' (intersection is product of probabilities)
Incorrectly identifying the denominator in conditional probability questions, often using the total sample space instead of the condition's probability
Failing to subtract the intersection when calculating the union, resulting in probabilities greater than 1
Confusing mutually exclusive events (where $P(A \cap B) = 0$) with independent events (where $P(A \cap B) = P(A)P(B)$)
In conditional probability questions, dividing by the probability of the numerator event rather than the probability of the condition (the 'given' event)
Failing to subtract the intersection $P(A \cap B)$ when calculating the union $P(A \cup B)$ for non-mutually exclusive events
Confusing mutually exclusive events (intersection is zero) with independent events (intersection is product of individual probabilities)
Incorrectly identifying the denominator in conditional probability questions, often using the total sample space (1) instead of the restricted sample space P(B)
Assuming independence without evidence and applying the multiplication rule P(A ∩ B) = P(A)P(B) inappropriately in general contexts
Failing to subtract the intersection P(A ∩ B) when calculating P(A ∪ B), leading to probabilities greater than 1
Writing answers in ratio notation (e.g., 3:4), which is treated as an incorrect method for probability by OCR examiners.
Failing to adjust the denominator in conditional probability questions (e.g., keeping the total as 10 instead of 9 for the second pick).
Incorrectly adding denominators when summing fractions (e.g., 1/5 + 2/5 = 3/10).
Confusing 'mutually exclusive' (cannot happen at same time) with 'independent' (one does not affect the other).

Key Terminology

Essential terms to know

The Probability Scale and Notation

Relative Frequency and Expectation

Mutually Exclusive and Exhaustive Events

Independent Events and the Multiplication Law

Tree Diagrams and Frequency Trees

Conditional Probability and Dependent Events

Venn Diagrams and Set Notation

The Probability Scale and Notation

Relative Frequency and Expectation

Mutually Exclusive and Exhaustive Events

Independent Events and the Multiplication Law

Tree Diagrams and Frequency Trees

Conditional Probability and Dependent Events

Venn Diagrams and Set Notation

Theoretical probability vs Relative Frequency

Mutually exclusive and Independent events

Tree diagrams and Venn diagrams

Conditional probability and Set notation

Sample space diagrams and systematic listing

Theoretical probability vs. Relative frequency

Mutually exclusive and Independent events

Sample space diagrams, Tree diagrams, and Venn diagrams

Conditional probability and Set notation

Theoretical probability and relative frequency

Mutually exclusive and independent events

Tree diagrams and Venn diagrams

Conditional probability and set notation

Expected frequency and risk analysis

Likely Command Words

How questions on this topic are typically asked

Calculate

Complete

Estimate

Show that

Work out

Interpret

Determine

Explain

Evaluate

Find

Criticise

Practical Links

Related required practicals

•{"code":"Modelling","title":"Medical Screening","relevance":"Application of conditional probability to determine false positive/negative rates"}
•{"code":"LDS","title":"Large Data Set","relevance":"Using relative frequency from the OCR Large Data Set to estimate probabilities for modeling"}
•{"code":"Large Data Set","title":"Sampling and Probability","relevance":"Application of probability distributions to samples drawn from the OCR Large Data Set"}

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