To calculate the conditional probability using a contingency table, you should divide the relevant cell’s value by the total for the specified condition.

A contingency table, often referred to as a two-way table, presents the frequency distribution of two or more categorical variables. To determine the conditional probability, concentrate on the specific condition outlined in your problem. For instance, if you want to find the probability of event $A$ occurring given that event $B$ has occurred, you should examine the row or column associated with event $B$.

Begin by identifying the total number of occurrences for the specified condition (event $B$). This total is typically obtained by summing the values within the row or column that corresponds to event $B$. Next, locate the cell that reflects the occurrence of both events $A$ and $B$. This cell indicates the joint frequency of events $A$ and $B$.

The conditional probability can then be computed by dividing the joint frequency (the cell value representing both $A$ and $B$) by the total frequency of the given condition (the sum of the row or column for event $B$). Mathematically, this can be expressed as:

For example, consider a contingency table that indicates there are $10$ students who like both mathematics and science, while $25$ students like science in total. The conditional probability that a student likes mathematics given that they like science can be calculated as follows:

This demonstrates how to effectively use a contingency table to find conditional probabilities.