Lesson 4

To get used to the definition, let us derive the following formula. Assuming that \(\Pr[B \cap C]>0\): \[\Pr[A \mid B \cap C] = \frac{\Pr[A \cap B \mid C]}{\Pr[B \mid C]}\ .\] We can derive it simply from the definition: \[\frac{\Pr[A \cap B \mid C]}{\Pr[B \mid C]}=\frac{\Pr[A \cap B \cap C]/\Pr[C]}{\Pr[B \cap C]/\Pr[C]}= \Pr[A \mid B \cap C] \ .\]

We can also prove the formula \(\Pr[A \mid B \cap C] = \frac{\Pr[A \cap B \mid C]}{\Pr[B \mid C]}\) visually: both sides of the equation are equal to the ratio of the areas of the dashed rectangles.