Joint, Conditional, and Marginal Probability for Contingency Tables

Suppose responses $(X,Y)$ of randomly chosen subjects have a probability distribution.

Joint probability

Let ${\pi }_{ij}$ be the probability that $(X,Y)$ occurs at row $i$ column $j$ in the contingency table. It is also referred to as the joint probability of $X$ and $Y$.

After sampling, let

• $n_{ij}$ : Cell frequencies
• $n$ : Total sample size

Then the sample joint probability is

$$p_{ij}=\frac{n_{ij}}{n}$$

Marginal probability

The marginal distributions are the row and column totals that result from summing the joint probabilities.

$$\begin{array}{rl}{\pi }_{i+}& =\sum _j{\pi }_{ij}\\ {\pi }_{+j}& =\sum _i{\pi }_{ij}\end{array}$$

Conditional probability

$${\pi }_{j|i}=\frac{{\pi }_{ij}}{{\pi }_{i+}}$$