The Life Starter
Sign up
Grade 11 · lesson 1 of 4

Read Joint and Marginal Counts

Build a two-way table and distinguish a joint event from a total category.

Download the grade pack

Supplied materials

  • Paper or a note app.
  • Fictional box: red circle 4, red noncircle 8, blue circle 6, blue noncircle 6. Each of the 24 cards is equally likely to be drawn.

Learn the idea

A two-way table classifies each card by two attributes. The cell for red and circle is a joint count: the card must satisfy both attributes. A row or column total includes all cards with just one named attribute, regardless of the other attribute.

For one unconditional draw, divide the relevant count by all twenty-four equally likely cards. Circle includes both red-circle and blue-circle cards. Counting only one circle cell would omit some allowed outcomes. Add all four cells to check the table total first.

  1. Draw rows red and blue and columns circle and noncircle; enter 4,8,6,6.
  2. Find row totals 12 and 12 and column totals 10 and 14; check the grand total 24.
  3. Calculate the chance of red and circle, then the chance of any circle.
Worked example

See the method in action

P(red and circle) = 4/24 = 1/6. P(circle) = (4 + 6)/24 = 10/24 = 5/12. The latter includes both colors in the circle column.

Your turn

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Using this same box, what is P(blue and circle) for one equally likely draw?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

1/4

The blue-circle cell contains six cards. The sample space is all 24 cards, so P(blue and circle) = 6/24 = 1/4.

Take it further

Move one red noncircle card into the red-circle cell. Counts become 5,7,6,6; the total stays 24. P(red and circle) becomes 5/24 and P(circle) becomes 11/24.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.