Conditional Probability and Two-Way Tables

Two-Way Tables

A two-way table shows frequencies for two categorical variables.

	Sports	No Sports	Total
Male	45	30	75
Female	35	40	75
Total	80	70	150

Conditional Probability

$P(A | B) = \frac{P(A \text{ and } B)}{P(B)} = \frac{\text{number in both A and B}}{\text{number in B}}$

The key: the denominator is the given condition's total, not the grand total.

Example: $P(\text{Sports} | \text{Male}) = \frac{45}{75} = 0.6$

Example: $P(\text{Male} | \text{Sports}) = \frac{45}{80} = 0.5625$

Notice these are different! The condition changes the denominator.

Relative Frequency

Joint relative frequency: each cell divided by the grand total. $P(\text{Male and Sports}) = \frac{45}{150} = 0.3$

Marginal relative frequency: row/column totals divided by the grand total. $P(\text{Male}) = \frac{75}{150} = 0.5$

Independence Test

Two events are independent if $P(A|B) = P(A)$.

Is playing sports independent of gender? $P(\text{Sports}) = \frac{80}{150} = 0.533$. $P(\text{Sports}|\text{Male}) = 0.6$.

Since $0.6 \neq 0.533$, sports and gender are NOT independent.

Tip 1: Identify the Denominator

The word after "given" (or the condition) determines the denominator. "Given male" → denominator is total males.

Tip 2: Read the Table Row/Column Carefully

Make sure you're reading the correct row and column.

Tip 3: "Among" or "Of" = Conditional

"Among males, what fraction play sports?" = $P(\text{Sports}|\text{Male})$.