Two-Way Tables

Calculating Probabilities

• Joint Probability: To find the probability of both events occurring, divide the joint frequency by the grand total: $P(A \text{ and } B) = \frac{\text{Joint Frequency}}{\text{Grand Total}}$.
• Marginal Probability: To find the probability of a single event occurring, divide the marginal total by the grand total: $P(A) = \frac{\text{Marginal Total}}{\text{Grand Total}}$.
• Conditional Probability: To find the probability of an event given a specific condition, divide the joint frequency by the marginal total of the condition: $P(A|B) = \frac{\text{Joint Frequency}}{\text{Marginal Total of B}}$.

Completing a Table

• Step 1: Fill in all known values provided in the problem description.
• Step 2: Use the row and column totals to solve for missing joint frequencies (e.g., $\text{Total} - \text{Known Cell} = \text{Unknown Cell}$).
• Step 3: Verify that the sum of row totals equals the sum of column totals, both matching the grand total.

• Frequency vs. Relative Frequency: A frequency is a raw count (e.g., 15 people), whereas a relative frequency is a proportion (e.g., 0.25 or 25%). Relative frequencies are essential for comparing datasets with different sample sizes.

• Identify the Denominator: This is the most common area for errors. If the question says 'Given that...' or 'Of the students who...', the denominator MUST be the marginal total for that specific group, not the grand total.

• The 'Total' Check: Always verify that your row totals and column totals add up to the same grand total. If they don't, there is an arithmetic error in your table.

• Read Carefully: Distinguish between 'and' (joint) and 'given' (conditional). 'And' looks at the whole group; 'given' looks at a specific row or column.

• Rounding: In probability questions, keep answers as fractions in simplest form unless the question specifically asks for decimals or percentages.

• Confusing Rows and Columns: Students often swap the 'condition' and the 'event' in conditional probability. Always identify which variable is the 'given' constraint first.

• Ignoring the Grand Total: When calculating marginal relative frequencies, students sometimes divide by a row total instead of the grand total.

• Assuming Independence: Never assume two variables are independent just because the table is symmetrical. You must calculate the conditional relative frequencies to prove independence.