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Problem 4. (20 points.) Do Q1.38 from Casella and Berger reproduced below:Prove each of the following statements. (Assume that any conditioning event has posi-tive probability.) 1. If P(B) = 1,

then P(A|B) = P(A) for any A. \text { 2. If } A \subset B \text {, then } P(B \mid A)=1 \text { and } P(A \mid B)=\frac{P(A)}{P(B)} \text {. } 3. If A and B are mutually exclusive, then P(A \mid A \cup B)=\frac{P(A)}{P(A)+P(B)}

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