USA - Geometry
Probability and statistics
Two-way frequency tables
20 practice questions
2 video lessons
Theory + worked examples
Theory
A two-way frequency table shows counts for two categories, with row and column totals. From it:
- Joint: a single cell divided by the grand total.
- Marginal: a row or column total divided by the grand total.
- Conditional: a cell divided by its row or column total.
The margins (totals) give marginal probabilities; the inner cells give joint probabilities.
A two-way table with row and column totals.
Reading probabilities from the table.
From counts in the table:
\[P(\text{joint})=\dfrac{\text{cell}}{\text{grand total}},\quad P(\text{cond.})=\dfrac{\text{cell}}{\text{row or column total}}\]
Divide by the grand total for joint/marginal, by a row or column total for conditional.
How to read a two-way table
- Find the relevant cell or total.
- Joint/marginal: divide by the grand total.
- Conditional: divide by the row or column total for the condition.
- Compare conditional and marginal to check for association.
Example 1 — Joint probability
From the table (\(50\) students), find \(P(\text{senior and sport})\).
Solution
Divide the cell count by the grand total.
| \(P\) | \(=\) | \(\dfrac{18}{50}=0.36\) |
Example 2 — Marginal probability
Find \(P(\text{plays a sport})\).
Solution
Use the column total over the grand total.
| \(P\) | \(=\) | \(\dfrac{30}{50}=0.6\) |
Example 3 — Conditional probability
Find \(P(\text{senior}\mid\text{sport})\).
Solution
Condition on the \(30\) who play a sport.
| \(P\) | \(=\) | \(\dfrac{18}{30}=0.6\) |
Example 4 — Check independence
Are “senior” and “sport” independent?
Solution
Compare \(P(\text{senior}\mid\text{sport})\) with \(P(\text{senior})\).
| \(P(\text{senior}\mid\text{sport})\) | \(=\) | \(0.6\) |
| \(P(\text{senior})\) | \(=\) | \(\dfrac{25}{50}=0.5\) |
\(0.6\neq0.5\), so they are not independent.
Common pitfalls
Joint uses the grand total; conditional uses a row or column total.
Condition on the correct margin — row vs column matters.
Totals must add up; check the margins before computing.
Frequently asked questions
What is a two-way frequency table?
A table of counts organized by two categories, with row and column totals.
What is a joint frequency?
The count in a single cell; its probability is the cell over the grand total.
What is a marginal frequency?
A row or column total; its probability is that total over the grand total.
How do you get a conditional probability from a table?
Divide a cell by its row or column total, not the grand total.
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