Two-way tables for probability
Two-Way Tables for Probability
Two-Way Tables for Probability is a topic in Statistics & Probability in the Common Core State Standards. It is aligned to Standard S-CP.4, which requires students to construct and interpret two-way frequency tables and use them to decide if events are independent and to approximate conditional probabilities.
A two-way frequency table organizes counts by two categories, giving joint, marginal, and conditional probabilities and a test for independence.
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.
From the counts:
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.
Cell over the grand total.
| \(\dfrac{18}{50}\) | \(=\) | \(0.36\) |
Column total over the grand total.
| \(\dfrac{30}{50}\) | \(=\) | \(0.6\) |
Condition on the \(30\) who play a sport.
| \(\dfrac{18}{30}\) | \(=\) | \(0.6\) |
Compare \(P(\text{senior}\mid\text{sport})=0.6\) with \(P(\text{senior})=0.5\).
| \(0.6\) | \(\neq\) | \(0.5\) |
Not independent.
Common pitfalls
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.