Algebra 2
Statistics and probability
Two-way tables for probability
20 practice questions
0 video lessons
Theory + worked examples
Theory
A two-way frequency table shows counts for two categories with row and column totals:
- Joint: a cell divided by the grand total.
- Marginal: a row/column total over the grand total.
- Conditional: a cell divided by its row or column total.
Compare a conditional with a marginal to test independence.
A two-way table with totals.
Reading probabilities from the table.
From the counts:
\[P(\text{joint})=\dfrac{\text{cell}}{\text{grand total}},\quad P(\text{cond.})=\dfrac{\text{cell}}{\text{row/col total}}\]
Marginals are the totals in the margins.
How to read the table
- Locate the relevant cell or total.
- Joint/marginal: divide by the grand total.
- Conditional: divide by the row or column total.
- Compare to test independence.
Example 1 — Joint probability
From the table (\(50\) students), find \(P(\text{senior and sport})\).
Solution
Cell over the grand total.
| \(\dfrac{18}{50}\) | \(=\) | \(0.36\) |
Example 2 — Marginal probability
Find \(P(\text{plays a sport})\).
Solution
Column total over the grand total.
| \(\dfrac{30}{50}\) | \(=\) | \(0.6\) |
Example 3 — Conditional probability
Find \(P(\text{senior}\mid\text{sport})\).
Solution
Condition on the \(30\) who play a sport.
| \(\dfrac{18}{30}\) | \(=\) | \(0.6\) |
Example 4 — Independence check
Are “senior” and “sport” independent?
Solution
Compare \(P(\text{senior}\mid\text{sport})=0.6\) with \(P(\text{senior})=0.5\).
| \(0.6\) | \(\neq\) | \(0.5\) |
Not independent.
Common pitfalls
Joint uses the grand total; conditional uses a row/column total.
Condition on the correct margin (row vs column).
Totals must add up — check the margins.
Frequently asked questions
What is a two-way frequency table?
A table of counts by two categories with row and column totals.
What is a joint probability?
A single cell divided by the grand total.
What is a conditional probability from the table?
A cell divided by its row or column total.
How do you test independence?
Compare a conditional probability with the matching marginal.
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