Contingency Tables: Counts, Marginal Totals, and Probabilities

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What is a contingency table?

A table that organizes observations according to categories of two variables, showing their combinations in cells and often displaying row, column, and grand totals.

What is a marginal total?

A row or column total at the edge of a contingency table. It counts observations in one category of a variable without separating them by the other variable.

What is a joint probability?

The probability that two events occur together. In a count table, find it by dividing the shared cell count by the grand total.

How do you calculate a conditional probability from counts?

Divide the count that satisfies both conditions by the total count in the group specified after “given.” For example, $P(\text{Studied in School} \mid \text{Woman})=38/72$.

How do you calculate a joint probability for sequential events?

Multiply the probability of the first event by the conditional probability of the second event given the first: $P(A \cap B)=P(A)P(B \mid A)$.

Quiz questions

1. In a contingency table of counts, what does an interior cell represent?Answer: The count for observations in both its row and column categories.
2. A sample contains 130 musicians, including 72 women. What is the probability that a randomly selected musician is a woman?Answer: $72/130$
3. Among 130 musicians, 38 are women who studied in school. What is the probability that a randomly selected musician is both a woman and studied in school?Answer: $38/130$
4. Among 72 women musicians, 38 studied in school. What is the probability that a musician studied in school given that the musician is a woman?Answer: $38/72$
5. A mouse chooses Door Three with probability $1/3$ and is caught given Door Three with probability $1/2$. What is the probability that it chooses Door Three and is caught?Answer: $(1/3)(1/2)=1/6$