\(\renewcommand{\P}{\mathbb{P}}\)

Conditional Probability Experiment

Sample Space
Event Prob Rel Freq
\( A \)
\( B \)
\( A^c \)
\( B^c \)
\( A \mid B \)
\( B \mid A \)
\( A^c \mid B \)
\( B^c \mid A \)
\( A \mid B^c \)
\( B \mid A^c \)
\( A^c \mid B^c \)
\( B^c \mid A^c \)

Description

In this app, \(A\) and \(B\) are rectangular events in a rectangular sample sapce \(S\). The sample space \( S \) is given the uniform distribution, so \( \P(E) = \text{area}(E) \big/ \text{area}(S) \) for every event \( E \). Events \( A \) and \( B \) can be moved by dragging an interior point of the rectangle, or resized by dragging the lower right corner of the rectangle. The list gives the events and their complements, and various events conditioned on other events. Of course, an ordinary probability can be thought of as a conditional probability, with the sample space \( S \) as the given event. When you click on an item in the list, the effective sample space is the region with either the light or dark shading, while the effective event is the region with the darker shading. When the experiment runs, the outcomes (elements of \( S \)) are shown as red dots. The relative frequency of each event is given in the table.