To Probability And Statistics By Rohatgi Zip [patched] | Solution Manual An Introduction

Case 1: ( P(A) = 0 ). Then ( P(A \cap B) \leq P(A) = 0 ) so ( P(A \cap B)=0 ). Thus ( P(A \cap B) = 0 = 0 \cdot P(B) = P(A)P(B) ). Case 2: ( P(A)=1 ). Then ( P(A^c)=0 ). Using Case 1, ( A^c ) and B are independent. Hence ( P(A \cap B) = P(B) - P(A^c \cap B) = P(B) - 0 = 1\cdot P(B) = P(A)P(B) ). QED.

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