Marginal Probabilities Use the Table’s Edges
In Using Two-Way Tables to Find Probabilities, you learned that a joint probability comes from an interior cell and a marginal probability comes from a row or column total. This tutorial focuses on reading those totals accurately and choosing a denominator that matches the selection described in the question.
A marginal probability concerns one variable alone. For instance, a table might classify people by both age group and composting habits. The probability of selecting someone in a particular age group does not depend on which composting category they belong to. Add the counts across that age-group row to include every person in the category.
The word “marginal” refers to the totals shown at the margins, or edges, of a table. A row total combines all the counts across that row; a column total combines all the counts down that column. Either kind of total can provide the count for a marginal probability.
The denominator is not chosen merely because a particular number appears nearby in the table. It depends on the group from which the person is selected. If the selection is from everyone represented, use the grand total. If a question explicitly says the selection is from a specified subgroup only, the selection pool has changed; do not report that as the marginal probability for the full table.
Read the Row or Column Before Dividing
Consider this invented survey of 360 community-garden members. Each member is classified by age group and whether they compost at home. The totals are included so you can see how the edge counts are formed.
| Age group | Composts at home | Does not compost | Total |
|---|---|---|---|
| Under 30 | 48 | 32 | 80 |
| 30–49 | 72 | 48 | 120 |
| 50 or older | 96 | 64 | 160 |
| Total | 216 | 144 | 360 |
The row total for people under 30 is \(48+32=80\). It includes both composting categories. The column total for people who compost is \(48+72+96=216\). It includes all three age groups. The grand total is 360, whether checked by adding the row totals or by adding the column totals.
Worked Example: Probability of a Row Category
One member is selected at random from all 360 members. Find the probability that the member is under 30, regardless of whether they compost.
Identify the event: The question asks only for age group. The member may be in either composting category, so use the “Under 30” row total, 80, rather than either interior cell.
Choose the denominator: The member is selected from all 360 members represented in the table. Therefore, use the grand total, 360.
Conclude: The probability that a randomly selected member from these 360 is under 30 is approximately \(0.2222\), or 22.22%. This is a marginal probability because it concerns age group without specifying composting status.
A quick check is that the two counts in the row add to its total: \(48+32=80\). If you used 48, you would answer the narrower question of being under 30 and composting, not the question as asked.
Worked Example: Probability of a Column Category
Using the same table, find the probability that a member selected at random from all 360 composts at home.
Identify the event: The event is “composts at home,” with no age group specified. Add the composting counts down that column: \(48+72+96=216\). This is the composting column total.
Choose the denominator: The selection is again from all 360 members, so divide by 360, not by one of the age-group totals.
Conclude: The probability that a randomly selected member from these 360 composts at home is \(0.60\), or 60%. This is a marginal probability for the composting variable.
The total for members who do not compost is 144, so the marginal probabilities for the two composting categories are \(216/360=0.60\) and \(144/360=0.40\). They add to 1, as expected, because every member is in exactly one of these two categories.
How to Avoid Dividing by the Wrong Total
Before calculating, say what the denominator represents in words. If the question says “select one member at random from the 360 members,” the denominator is 360. If it says “select one member at random from those who compost,” the selection pool is the 216 composters instead. Those are different questions: the first asks about a category’s share of the full group, while the second restricts attention to a subgroup.
A row or column total is usually the numerator for a marginal probability from the full table. It is not the denominator merely because it belongs to the category named in the question. Using a row total as the denominator changes the pool to that row. This can be appropriate only when the question explicitly restricts the selection to that row.
Worked Example: Spot the Denominator Change
An invented transportation survey of 240 residents records each person’s usual way of getting to a neighborhood center and whether they live in the north or south part of town.
| Usual travel method | North | South | Total |
|---|---|---|---|
| Bus | 54 | 36 | 90 |
| Bicycle | 30 | 42 | 72 |
| Walk | 24 | 54 | 78 |
| Total | 108 | 132 | 240 |
Question A: A resident is selected at random from all 240 surveyed residents. What is the probability that the resident usually bicycles?
The bicycle row total is \(30+42=72\). Since the selection is from all 240 residents, divide by 240:
The probability that a randomly selected resident from the full survey group usually bicycles is \(0.30\), or 30%.
Question B: Now suppose a resident is selected at random from the 108 north-side residents only. What fraction of that group usually bicycles?
The event count is now the bicycle-and-north cell, 30. The selection pool is only the north-side residents, so the denominator is 108:
About 27.78% of the north-side residents in this table usually bicycle. This is not the marginal probability of bicycling in the full group. The numerator and denominator both reflect the restricted north-side selection.
Compare: For the full-group question, \(72/240=0.30\). For the north-side-only question, \(30/108\approx0.2778\). Neither calculation is an arithmetic error: each answers a different question because the selection pool differs.
Use Totals to Check the Probability
A two-way table provides useful checks before and after calculation. First, each row total should equal the sum of the interior cells in that row, and each column total should equal the sum of its interior cells. Second, the row totals should add to the grand total, and the column totals should also add to it.
For the transportation table, the row totals are \(90,72,\) and \(78\), and \(90+72+78=240\). The column totals are 108 and 132, and \(108+132=240\). The marginal probabilities of the three travel methods are \(90/240=0.375\), \(72/240=0.30\), and \(78/240=0.325\); together, they add to 1.
This last check works because the categories of one variable account for everyone in the table exactly once. For example, each resident has one recorded usual travel method: bus, bicycle, or walking. If the probabilities for those categories do not add to 1, check whether a total or denominator was misread or calculated incorrectly.
Common Mistakes and AP Exam Tips
- Using an interior cell for a category regardless of the other variable. If the question asks for everyone who bicycles, regardless of location, use the bicycle row total, not just the north-and-bicycle or south-and-bicycle cell.
- Dividing by the category total automatically. For the probability of bicycling from all 240 residents, \(72/240\) is appropriate. Dividing by 72 would use the bicyclists as the selection pool, not all residents.
- Mixing up the numerator and denominator. The marginal total counts people who meet the event description; the denominator counts the full selection pool stated in the question.
- Ignoring phrases such as “from the north-side residents.” Such wording changes the selection pool. Identify it before choosing the denominator.
- Giving only a decimal without context. State what the probability describes, identify the relevant total, and show the division.
For a clear, full-credit explanation, name the event, point to its row or column total, state the selection pool, show the fraction, and interpret the result in context. In particular, make clear why the denominator is the grand total or, when specified, a subgroup total.
Check Your Understanding
Use the transportation table of 240 residents above. Answer each question and explain which total supplies the denominator.
- What is the probability that a resident selected from all 240 usually walks?
- What is the probability that a resident selected from all 240 lives in the south part of town?
- Why is 30 not the numerator for the marginal probability of usually bicycling?
- Among the 108 north-side residents only, what fraction usually walks?
- Explain why the probabilities for bus, bicycle, and walking should add to 1 when each resident records exactly one usual travel method.