In this lesson:
By the end of this pair, you should be able to:
You survey students, recording two things about each:
A plain list of 120 students can't answer questions fast. We need a table.
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.
The margins must check out before any probability:
If the two sums don't match, a cell or total is wrong. Reconcile first.
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.
Once built, treat the table as a sample space:
Every probability is a count ÷ grand total. The table becomes a probability model.
For a randomly selected student:
Both divide by the grand total. Keep that denominator fixed today.
Use the science column total over the grand total:
"Marginal" = one category over the whole sample.
Use a single cell over the grand total:
Joint isolates one cell; the denominator is still the grand total.
Different numerators, same denominator. Conditional probability changes the denominator — that's next lesson.
From the favorite-subject table, find and label each:
Write each as a fraction over the grand total, then label its type.
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.
✓ 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.
Click to begin the narrated lesson
Construct and interpret two-way frequency tables