Two-way tables organize data for two categorical variables, such as study method and exam result. They let you see patterns that are hard to notice in a long list of responses. From one table, you can compute joint, marginal, and conditional distributions.
These probabilities help you compare groups fairly and support evidence-based conclusions.
Key Facts
- Joint probability: P(A and B) = count in cell / grand total.
- Marginal probability: P(A) = row total or column total / grand total.
- Conditional probability: P(A | B) = count in both A and B / total for B.
- Multiplication rule: P(A and B) = P(A | B)P(B).
- Independence check: A and B are independent if P(A | B) = P(A).
- For every row or column conditional distribution, the conditional probabilities should add to 1.
Vocabulary
- Joint distribution
- A display of probabilities or relative frequencies for all combinations of two categorical variables.
- Marginal distribution
- A distribution that uses only the row totals or column totals of a two-way table.
- Conditional distribution
- A distribution of one variable calculated only within a selected category of another variable.
- Grand total
- The total number of observations in the entire two-way table.
- Independence
- Two events or variables are independent when knowing one does not change the probability of the other.
Common Mistakes to Avoid
- Using the grand total for every probability is wrong because conditional probabilities use the total of the condition group as the denominator.
- Mixing up P(A | B) and P(B | A) is wrong because the condition determines which row or column total goes in the denominator.
- Calling a cell count a probability is wrong because probabilities must be divided by the appropriate total and usually range from 0 to 1.
- Checking independence with counts alone is wrong because groups may have different sizes, so you must compare probabilities such as P(A | B) and P(A).
Practice Questions
- 1 A survey of 100 students records whether they exercise and whether they sleep at least 8 hours. The table counts are: Exercise and 8+ hours = 30, Exercise and less than 8 hours = 20, No exercise and 8+ hours = 10, No exercise and less than 8 hours = 40. Find P(exercise and 8+ hours), P(8+ hours), and P(8+ hours | exercise).
- 2 In a class of 80 students, 50 passed a test and 30 did not. Of the 50 who passed, 35 used a study guide. Of the 30 who did not pass, 12 used a study guide. Find P(used study guide | passed) and P(passed | used study guide).
- 3 A two-way table shows that P(likes math) = 0.60, but P(likes math | plays a musical instrument) = 0.60 as well. Explain what this suggests about the relationship between liking math and playing a musical instrument.