Venn Diagrams and Probability

You need Java to see this.

Venn diagrams are a way of visualizing sets. This applet lets you highlight
the set A, the set B, the universal set S, the empty set {}, the complement of A
(Ac), the complement of B (Bc), the intersection of A and B (AB), the union of A
and B (A or B), the intersection of the complement of A with B (AcB), and the
intersection of A with the complement of B (ABc). One can think of the relative
area of a set to the area of S as a probability. The picture uses that analogy:
the numbers P(A) and P(B) are proportional to the area of A and the area of B,
respectively; P(AB) is the area of the intersection of A and B, and P(A or B) is
the area of the union of A and B.

You can drag the events
A and B around, which changes the probability of their intersection and of their
union. When A and B do not overlap, they are said to be disjoint
or mutually exclusive. When the probability of the intersection of A and B
equals the product of the probability of A and the probability of B (when P(AB)
= P(A)×P(B)), A and B are independent.
Try to drag A or B to make P(AB) = P(A)×P(B); notice how hard it is to make the
overlap just right: independence is a very special relationship between events.
The scrollbars let you adjust the probability of A and the probability of B.