Contingency table

The same information as the Venn diagram can be arranged in a $2\times 2$ table, where the row and column sums automatically give the marginal probabilities:

	$A$	$\overline{A}$	Total
$B$	$y$	$z$	$\mathbf{P(B)=y+z}$
$\overline{B}$	$x$	$t$	$\mathbf{P(\overline{B})=x+t}$
Total	$\mathbf{P(A)=x+y}$	$\mathbf{P(\overline{A})=z+t}$	$\mathbf{1}$

• The four inner cells are the joint probabilities $P(A\cap B)$, $P(\overline{A}\cap B)$, $P(A\cap\overline{B})$, $P(\overline{A}\cap\overline{B})$.
• The row and column totals are the marginal probabilities.
• The overall sum is always $1$.

Observation

The contingency table is particularly handy when we know percentages of the type “who has characteristic $A$ among those who have $B$” (that is, a conditional probability): it is enough to fill in one cell per row and obtain the others through the sums.

Links

Topics: Probability
Concepts: Conditional probability · Joint probability · Marginal probability · Contingency table
Methods: Contingency table
Skills: Probability calculation