Build and Read a Two-Way Table

Lesson 1 of 2: Construction and Probabilities

In this lesson:

• Construct a two-way frequency table from raw data
• Read joint and marginal probabilities

Goals for This Two-Lesson Pair

By the end of this pair, you should be able to:

1. Construct a two-way frequency table with all totals
2. Read joint and marginal probabilities
3. Approximate conditional probabilities by restricting
4. Decide whether two events are independent
5. Interpret comparisons, separating association from causation

Two Labels per Student — Organize Them

You survey students, recording two things about each:

• Their grade level (10th or 11th)
• Their favorite subject (math, science, English)

A plain list of 120 students can't answer questions fast. We need a table.

Tally Into Cells, Total the Margins

Rows = grade, columns = subject. Tally each student into one cell.

Interior cells hold joint counts; the edges hold margins; the corner holds the grand total.

Name the Parts of the Table

• Joint cell: one combination (24 = 10th + science)
• Marginal total: one category (40 favor science)
• Grand total: the whole sample (120)

Reconcile the Margins Before Computing

The margins must check out before any probability:

• Row totals: ✓
• Column totals: ✓

If the two sums don't match, a cell or total is wrong. Reconcile first.

Quick Check: Do the Margins Reconcile?

Row totals are 35 and 45; the grand total reads 90.

Do they reconcile? Decide before advancing.

Answer: — no, a cell or total is wrong.

The Table Is a Sample Space

Once built, treat the table as a sample space:

• Picking a random student = an outcome
• Any group of cells = an event

Every probability is a count ÷ grand total. The table becomes a probability model.

Any Count Over the Grand Total

For a randomly selected student:

• A single cell → a joint probability
• A row or column total → a marginal probability

Both divide by the grand total. Keep that denominator fixed today.

A Marginal Probability: Favors Science

Use the science column total over the grand total:

• Lumps the whole science column together, ignoring grade
• Denominator is the grand total, 120

"Marginal" = one category over the whole sample.

Joint Probability: Science and Tenth Grade

Use a single cell over the grand total:

Joint isolates one cell; the denominator is still the grand total.

Marginal Lumps and Joint Isolates

Different numerators, same denominator. Conditional probability changes the denominator — that's next lesson.

Your Turn: Compute and Label

From the favorite-subject table, find and label each:

1. — marginal or joint?
2. — marginal or joint?

Write each as a fraction over the grand total, then label its type.

Watch Out: Two Habits to Lock In

Reconcile first: row totals and column totals must each sum to the grand total.

Marginal vs joint: whole row/column vs single cell — same grand-total denominator.

Get these right and conditional probability comes easily.

Key Takeaways and What's Next

✓ Build the table, then reconcile the margins
✓ The table is a sample space: count ÷ grand total
✓ Marginal = row/column total; joint = single cell

Every denominator here is the grand total

Next: restrict to a row for conditional probability, then decide independence.