Two-Way Table vs Conditional Probability

A two-way table, also called a contingency table, cross-classifies each individual by two categorical variables to show where the categories overlap. Row and column totals, called marginal totals, let you compute conditional and joint proportions to compare groups. For example, a table of 100 students by grade level in rows and whether they play a sport in columns reveals the sport rate within each grade. Such a table is the input to a chi-square test of independence or homogeneity, through its expected counts.

Conditional probability updates the chance of an event once you know another event has happened. It is written $P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$ (the probability of $A$ given $B$), the probability of both divided by the probability of the given event. For example, if 40 percent of students take both Spanish and art and 50 percent take Spanish, then the chance of art given Spanish is $0.40 / 0.50 = 0.8$. Restricting attention to $B$ is what given means.