Two-way tables for outcomes of two events

Probability · Outcome tables

Where does each outcome live?

A win/lose spinner is spun twice. The outcome table is a 2 by 2 grid, and every cell is where one column meets one row. Pick an outcome, then pick the cell it belongs in. Then use the buttons to turn the table round.

First spin = columns, second spin = rows. Each spin picks one side of the table, and the cell is where they meet.

4 of 4 still to sort.

Same four outcomes in both tables. Only their positions move.

Predict, then check

Picture a 5 by 5 grid. The first spin goes across the top, the second goes down the side, and each cell shows the two numbers added together.

A spinner numbered 1 to 5 is spun twice. The totals run from 2 up to 10. Would a total of 7 and a total of 10 be equally likely?

What you need to know

• A two-way table lists every outcome of a two-stage trial, and the outcomes inside the grid are called the sample space.
• One stage goes across the columns, the other down the rows: columns count the first stage's outcomes, rows the second's.
• Each cell holds only the outcome for its own column and its own row. An outcome in the wrong cell is wrong.
• Have a goSam is very sure. A star/moon spinner is spun, then a 1/2/3 spinner. Sam gets 'moon, then 3', with the first spin across the top, and writes it in the '3' column and the 'moon' row. Is Sam right?

  No. Moon was the first spin, so it picks the column. The 3 picks the row. It goes in the moon column, row 3.

  Each spin picks one side of the table and the cell is where they meet. Sam has swapped the jobs of the two spins.

• Swap the rows and columns and it's the same table, just turned round.
• Whether AB and BA are the same outcome depends on the context: sometimes they are, sometimes they aren't.
• Totals from two spins aren't equally likely: on a 1–5 spinner, 7 appears four times in the table, 10 only once.
• For three stages, tabulate two stages, treat those outcomes as one new spinner, then tabulate them against the third stage.
• Have a goA first table has 2 columns and 3 rows. You treat its outcomes as one new spinner. How many sections does that new spinner have?

  6 sections, one for each cell of the table.

  Every cell of the first table holds one outcome, so the new spinner has as many outcomes as the table has cells.

• You can also sort a trial's outcomes by two events: each outcome goes in the one cell that meets both conditions.
• Have a goA spinner is numbered 1 to 6 with the events 'odd' and 'more than 3'. Does the outcome 5 meet both events, just one, or neither?

  Both. 5 is odd and it is more than 3.

  An outcome goes in the one cell that meets the conditions of both events, and 5 passes both tests.

• In a frequency table, a cell counts items fitting both categories; a row or column total counts one category.
• If two values in a row or column are known, you can find the missing third. Check the final value both ways.

The big picture

A two-way outcome table puts one stage across the columns and the other down the rows, so each cell is the one outcome where its row and column meet. It lists every outcome, so it also gives the sample space. It can total two spins, sort outcomes by two events and extend to three stages. Two-way frequency tables use totals to find missing values.

⚠ Watch out

Dropping an outcome into a cell without checking BOTH its column and its row, or thinking that swapping rows and columns gives a different table. A cell only ever holds the outcome that matches both, and a turned-round table is the same table.

Memory hook

A cell is a meeting point. It holds only the outcome that fits its column AND its row.

Check yourself

One spinner has red, blue, green; another has 1, 2. Colours across the top: how many columns and rows, and which cell holds 'green, then 2'? (3 columns, 2 rows, green column meets row 2.)