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All courses Geometry · S-CP.4 43 of 57
Construct and interpret two-way frequency tables as sample spaces for independence and conditional probability.

Complete a two-way frequency table, including all margins

Problem
Complete the displayed grade-and-tea table by summing each grade row, each tea column, and the grand total. Check that \(\text{both}~\text{margin}~\text{sums}\) agree.
Prompt two-way table. Rows are grade 9 and grade 10; columns are likes tea and does not like tea. The four cells are 12, 8, 10, and 20 in row order. All row totals, column totals, and the grand total are unknown. Open full size
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Hint

Place the four given counts first, then sum across each row and down each column.

Each interior count belongs at one row-column intersection. Row and column totals are sums, not additional observations.

Solution walkthrough

01

Add each grade row

\[\text{grade}~9:~12+8=20;~\text{grade}~10:~10+20=30\]

Each row total combines tea and no-tea counts for that grade.

02

Add each tea column

\[\text{tea}:~12+10=22;~\text{no}~\text{tea}:~8+20=28\]

Each column total combines both grade levels for one response.

03

Compute the grand total two ways

\[\text{row}~\text{totals}:~20+30=50;~\text{column}~\text{totals}:~22+28=50\]

Agreement of the independent margin sums checks all four totals.

04

Report the completed margins

\[\text{grade}~9=20;~\text{grade}~10=30;~\text{tea}=22;~\text{no}~\text{tea}=28;~\text{grand}=50\]

Every missing cell is filled and both required checks equal 50.

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Another way

  1. Sum all four interior cells directly: 12+8+10+20=50.

!

Common mistake

Do not swap row totals with column totals; keep each label attached to the cells it summarizes.

Completed table. Grade 9 total 20; grade 10 total 30; likes-tea total 22; does-not-like-tea total 28; grand total 50.
Highlighted cells show the requested event and its correct reference total.

Solution walkthrough video