Two Way Tables

Joint Probability: The likelihood of an individual belonging to two specific categories simultaneously, calculated as $P(A \text{ and } B) = \frac{\text{Cell Frequency}}{\text{Grand Total}}$.

Marginal Probability: The likelihood of an individual belonging to a single category, calculated as $P(A) = \frac{\text{Row or Column Total}}{\text{Grand Total}}$.

Conditional Probability: The likelihood of an event occurring given that another condition is already met, calculated as $P(A|B) = \frac{\text{Joint Frequency of A and B}}{\text{Total of Category B}}$.

Mathematical Definition: Two events $A$ and $B$ are considered independent if the occurrence of one does not affect the probability of the other.

The Multiplication Rule: In a two-way table, independence is verified if the joint probability equals the product of the marginal probabilities: $P(A \cap B) = P(A) \times P(B)$.

Alternative Check: Independence can also be confirmed if the conditional probability equals the marginal probability, such as $P(A|B) = P(A)$.

Verify Totals First: Before answering any questions, always sum the rows and columns to ensure they match the grand total; a single arithmetic error here will cascade through all subsequent calculations.

Identify the 'Given': In word problems, look for phrases like 'given that', 'if', or 'of the [category]'; these indicate a conditional probability where the denominator must be a marginal total, not the grand total.

Independence Proofs: When asked to determine independence, always show the calculation for $P(A) \times P(B)$ and compare it explicitly to the value of $P(A \cap B)$ from the table.

Reasonableness Check: Probabilities must always be between $0$ and $1$; if your calculation results in a number greater than $1$, you likely swapped the numerator and denominator.