Conditional probability — A/L Combined Mathematics

What must I remember about Conditional probability?

• Conditional Probability Definition: The probability of event A occurring given that event B has ALREADY occurred is P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.
• Sample space reduction: Conditioning on B effectively shrinks the relevant sample space from S to B.
• Multiplication Rule of Probability: P(A ∩ B) = P(B) * P(A | B) = P(A) * P(B | A); for three events: P(A ∩ B ∩ C) = P(A) * P(B | A) * P(C | A ∩ B).
• Test for Independence via conditional probability: Events A and B are independent if and only if P(A | B) = P(A) (or P(B | A) = P(B)).
• Tree diagrams for conditional paths: Branch probabilities along paths multiply; disjoint branch terminal outcomes add.

Test yourself (1 minute)

1. P(A | B) ≠ P(B | A) asymmetry

   Conditional probability is strictly asymmetric; e.g., P(cloudy | raining) = 1, but P(raining | cloudy) is far less than 1.

2. With vs Without Replacement in urn models

   Drawing balls without replacement makes successive draws DEPENDENT (probabilities change); drawing with replacement preserves INDEPENDENCE (probabilities remain constant).

3. Law of Total Probability

   If B1, B2, ..., Bn form a partition of S (mutually exclusive and exhaustive): P(A) = ∑_i=1ⁿ P(Bi) * P(A | Bi).

Practise Conditional probability questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 7

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Where does Conditional probability fit in the syllabus?

Year 2 · Term 3