Exercise C.2.4

Prove that

$$ \Pr\{A | B\} + \Pr\{\overline A | B\} = 1 $$

Very obvious if you think about it, but anyway:

$$ \begin{aligned} \Pr\{A|B\} + \Pr\{\overline A | B\} &= \frac{\Pr\{A \cap B\}}{\Pr\{B\}} + \frac{\Pr\{\overline A \cap B\}}{\Pr\{B\}} \\ &= \frac{\Pr\{A \cap B\} + \Pr\{\overline A \cap B\}}{\Pr\{B\}} \\ &= \frac{\Pr\{(\overline A \cap B) \cup (A \cap B)\}}{\Pr\{B\}} \\ &= \frac{\Pr\{(A \cup \overline A) \cap B)}{\Pr\{B\}} \\ &= \frac{\Pr\{B\}}{\Pr\{B\}} \\ &= 1 \end{aligned} $$