Conditional probability — A/L Combined Mathematics

Conditional probability measures the probability of an event A occurring given that another event B has already occurred with P(B) > 0, defined as: P(A | B) = P(A ∩ B) / P(B). This restricts the effective sample space to B. From this definition follows the Multiplication Rule of Probability: P(A ∩ B) = P(B) P(A | B) = P(A) P(B | A). Two events A and B are statistically independent if and only if knowledge of B gives no information about A: P(A | B) = P(A), which strictly reduces the multiplication rule to the necessary and sufficient condition: P(A ∩ B) = P(A) P(B).

Check before you answer (A/L Applied Maths Paper 2)

1. Denominator Event Identification: Correctly identifying the given conditioned event B and evaluating its probability P(B) as the denominator.
2. Multiplication Rule with Dependent Events: Adjusting successive branch probabilities in sampling without replacement (e.g., drawing cards or colored marbles).
3. Independence Proof Rigour: Testing whether P(A ∩ B) equals P(A) × P(B) explicitly; never assuming independence unless explicitly stated.