Probability and Conditional Modelling
Using set notation, diagrams, addition and multiplication rules, conditional probability, independence and discrete probability models.
How to study A-level Mathematics
Define the objects and conditions, select a representation, execute a justified method, then validate and interpret the result.
Core concepts
Concept 1
Probability can be organised through complements, mutually exclusive events, unions, intersections and conditional events.
Exam cue: Define events and translate words into set or conditional notation first.
Concept 2
Independence means one event does not change the probability of another and has a precise multiplicative test.
Exam cue: Use a tree, table or Venn diagram that preserves the conditioning structure.
Concept 3
A probability distribution assigns non-negative probabilities summing to one and supports expectation and variance.
Exam cue: Check every probability lies from zero to one and the total distribution is one.
Risk pitfalls and guardrails
Treating mutually exclusive events as independent.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Reversing the condition in a conditional probability.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Adding branch probabilities when the events occur in sequence.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Memory anchors
Complement
The complement of A has probability 1-P(A).
Mutually Exclusive
Mutually exclusive events cannot occur together.
Conditional Probability
P(A|B) is the probability of A after restricting to B.
Independence
Independent events satisfy P(A∩B)=P(A)P(B).
Expected Value
Expected value is the probability-weighted mean of a random variable.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
If P(A)=0.37, find P(Aᶜ).
Given P(A)=0.6, P(B)=0.5 and P(A∩B)=0.3, find P(A∪B).
Answer all questions to submit.
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Move forward only after this module is stable.
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