Two-Way Tables and Conditional Probability Cheat Sheet

Key Facts

• A two-way table shows counts for two categorical variables, with row totals, column totals, and a grand total.
• The joint probability of events $A$ and $B$ is $P(A\cap B)=\frac{\text{count in both }A\text{ and }B}{\text{grand total}}$.
• The marginal probability of event $A$ is $P(A)=\frac{\text{total count in }A}{\text{grand total}}$.
• The conditional probability of $A$ given $B$ is $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ when $P(B)>0$.
• Using counts from a table, $P(A\mid B)=\frac{\text{count in both }A\text{ and }B}{\text{total count in }B}$.
• The complement rule is $P(A^c)=1-P(A)$, where $A^c$ means the event $A$ does not happen.
• Events $A$ and $B$ are independent if $P(A\mid B)=P(A)$, meaning knowing $B$ does not change the probability of $A$.
• Events $A$ and $B$ are also independent if $P(A\cap B)=P(A)P(B)$.

Vocabulary

Two-Way Table

A table that organizes counts or frequencies for two categorical variables at the same time.

Joint Probability

The probability that two events both occur, written as $P(A\cap B)$.

Marginal Probability

The probability of one event found from a row total or column total divided by the grand total.

Conditional Probability

The probability that event $A$ occurs given that event $B$ has already occurred, written as $P(A\mid B)$.

Grand Total

The total number of observations in the entire two-way table.

Independence

A relationship where knowing that one event occurred does not change the probability of another event.