Practice Two-Way Tables in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A table that displays frequencies for two categorical variables simultaneously, organized with one variable in rows and the other in columns. It shows joint frequencies (individual cells), marginal frequencies (row/column totals), and enables calculation of conditional frequencies.

Imagine surveying students about their favorite sport AND their grade level. A two-way table is like a grid: grades go down the side, sports go across the top, and each cell tells you how many students are in that specific combination. The totals on the edges (margins) tell you the overall counts for each category.

Showing a random 20 of 50 problems.

Example 1

challenge
A two-way table: 240 people. 144 like soccer; 96 like basketball. Of soccer fans, 60 are kids. Of basketball fans, 48 are kids. Are 'sport choice' and 'kid vs adult' independent?

Example 2

medium
A survey of 80 students: 30 are 6th-graders, of whom 18 walk to school; 50 are 7th-graders, of whom 20 walk. What fraction of 6th-graders walk?

Example 3

medium
Same table as before: do the rates P(Yes∣Male)=0.6P(\text{Yes} \mid \text{Male}) = 0.6 and P(Yes∣Female)=0.3P(\text{Yes} \mid \text{Female}) = 0.3 suggest the variables are associated or independent?

Example 4

hard
A two-way table reports rounded percentages by ROW. Row 1: 60% Yes, 40% No. Row 2: 60% Yes, 40% No. Is the row variable a useful predictor of Yes/No?

Example 5

hard
A 2x2 table: row totals 50 and 50; column totals 40 and 60. Compute the EXPECTED count in the (row1, column1) cell under independence.

Example 6

medium
A two-way table: 100 people; 60 vegetarian, 40 non-vegetarian. 20 of the non-vegetarians chose salad. What is P(salad∣non-vegetarian)P(\text{salad} \mid \text{non-vegetarian})?

Example 7

medium
A two-way table: of 90 students, 54 passed. Of those who passed, 36 studied. What fraction of passers did NOT study?

Example 8

medium
Of 200 surveyed, 80 are adults; of those adults, 60 own a pet. What is the marginal proportion of adults who own a pet, divided by the grand total?

Example 9

easy
In a two-way table, what do the row and column totals represent?

Example 10

medium
A two-way table has a missing cell. Row total is 40; the other cell in that row is 15. What is the missing cell?

Example 11

medium
Two-way table has row totals 80 and 120, and column totals 90 and X. Find X and the grand total.

Example 12

medium
A two-way table: Cats(owners) 24, Dogs(owners) 36, total 60. Of dog owners, 27 also have a yard. What fraction of dog owners have a yard?

Example 13

easy
A two-way table: Left-handed/Male=12, Left-handed/Female=8, Right-handed/Male=88, Right-handed/Female=92. Find the marginal proportion of left-handers and P(Left-handed|Male).

Example 14

easy
A two-way table records two kinds of variables. What type of variables are they?

Example 15

challenge
A two-way table: Survey of 150. 90 like the app; of those, 54 are students. 60 dislike it; of those, 12 are students. What fraction of ALL students like the app?

Example 16

medium
A 2x2 table records 50 students: 30 in honors, 20 not. Of the honors students, 24 take a foreign language; of the non-honors, 12 take one. What fraction of foreign-language learners are honors students?

Example 17

challenge
In a two-way table, P(A given B)=3040P(\text{A given B})=\frac{30}{40} and P(A given not B)=3040P(\text{A given not B})=\frac{30}{40}. Are events A and B independent? Explain.

Example 18

challenge
A 2x2 table: (A,Yes)=20, (A,No)=30, (B,Yes)=10, (B,No)=40. Find the expected count for (A,Yes) under independence and compare to the observed.

Example 19

medium
A two-way table records 150 people: 90 like tea (60 of those are women) and 60 like coffee (20 of those are women). What fraction of women like tea?

Example 20

medium
Grand total 120. 'Owns a bike' column total is 72. Among bike-owners, 27 walk to school. What fraction of bike-owners walk?
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