If for two events [math: A] and [math: B] , [math: Pr(A)=920] Pr(A)= \frac{9}{20} , [math: Pr(B)=12] Pr(B)= \frac{1}{2} , and [math: Pr(A∩B)=15] Pr(A\cap B)= \frac{1}{5} , then [math: P(A|B)] is equal to :

Question

If for two events [math: A] and [math: B] , [math: Pr(A)=920] Pr(A)= \frac{9}{20} , [math: Pr(B)=12] Pr(B)= \frac{1}{2} , and [math: Pr(A∩B)=15] Pr(A\cap B)= \frac{1}{5} , then [math: P(A|B)] is equal to :

BlackTom AI · Accepted Answer

Let me restate the given information and then evaluate each option step by step.

Given: Pr(A) = 9/20, Pr(B) = 1/2, Pr(A ∩ B) = 1/5. We want P(A|B).

Recall the definition: P(A|B) = P(A ∩ B) / P(B).

- Option A suggests 9/40. If we test this, we would need P(A ∩ B) / P(B) = 9/40. Since P(B) = 1/2, this would imply P(A ∩ B) = (9/40) * (1/2) = 9/80, which contradicts the given P(A ∩ B) = 1/5. Therefore, A is incorrect.

- Option B suggests 9/20. Using P(A|B) = P(A ∩ B) / P(B) and P(B) = 1/2, this would require P(A ∩ B) = (9/20) * (1/2) = 9/40, which again does not match the given 1/5. Hence B is incorrect.

- Option C suggests 2/5. Compute directly: P(A|B) = P(A ∩ B) / P(B) = (1/5) / (1/2) = (1/5) * 2 = 2/5. This matches the value given in option C, so it is consistent with the provided data.

- Option D suggests 4/9. If P(A|B) were 4/9, then P(A ∩ B) would have to be (4/9) * P(B) = (4/9) * (1/2) = 4/18 = 2/9, which does not equal the given P(A ∩ B) = 1/5. Therefore D is incorrect.

Summary of the reasoning: evaluating P(A|B) via the definition shows the provided intersection and B-probability yield P(A|B) = (1/5) ÷ (1/2) = 2/5. The other options fail because their implied P(A ∩ B) values (or derived products) do not align with the given Pr(A ∩ B) = 1/5.

MUF0142 Fund. Mathematics Unit 2 - Semester 2, 2025

If for two events [math: A] and [math: B] , [math: Pr(A)=920] Pr(A)= \frac{9}{20} , [math: Pr(B)=12] Pr(B)= \frac{1}{2} , and [math: Pr(A∩B)=15] Pr(A\cap B)= \frac{1}{5} , then [math: P(A|B)] is equal to :

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