Four-region Venn diagram

Almost every probability problem with two events $A$ and $B$ can be handled with two equivalent tools: the (four-region) Venn diagram or the contingency table. Being able to move from one to the other is the winning move.

The sample space is a rectangle; the two events are two overlapping ovals. The four regions are called $x, y, z, t$:

The sample space $\Omega$ is the rectangle; the two events $A$ and $B$ are overlapping ovals that mark out four regions $x, y, z, t$.

The four regions represent the four joint probabilities:

• $x = P(A\cap\overline{B})$   (“only $A$”)
• $y = P(A\cap B)$   (“both”)
• $z = P(\overline{A}\cap B)$   (“only $B$”)
• $t = P(\overline{A}\cap\overline{B})$   (“neither $A$ nor $B$”)

The four quantities must satisfy the normalisation constraint:

$x + y + z + t = 1,$

and moreover from them the marginal probabilities and the probabilities of union and intersection are obtained:

$P(A) = x+y, \quad P(B) = y+z, \quad P(A\cap B) = y, \quad P(A\cup B) = x+y+z.$

Links

Topics: Probability
Concepts: Venn diagram · Joint probability · Marginal probability
Methods: Venn diagram prob
Skills: Calculating probability · Interpreting a graph