Question:
If $A$ and $B$ are any two events such that $P(A)+P(B)-P(A$ and $B)=P(A)$, then
(A) $P(B \mid A)=1$
(B) $P(A \mid B)=1$
(C) $P(B \mid A)=0$
(D) $P(A \mid B)=0$
Solution:
$P(A)+P(B)-P(A$ and $B)=P(A)$
$\Rightarrow \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A})$
$\Rightarrow \mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$
$\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{B})$
$\therefore P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{P(B)}{P(B)}=1$
Thus, the correct answer is B.
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