One Respondent Can Belong to More Than One Category
The previous tutorial, Probability of At Most and At Least in Discrete Categories, added probabilities for categories that cannot happen together. That works when the categories are disjoint: one outcome cannot belong to two of them at once. Survey answers need a careful check before using that addition rule. A person may be able to select several answers, so one respondent can belong to more than one response category.
For instance, a survey might ask which ways someone traveled to school last week. A respondent who took a bus on Monday and rode a bike on Friday could reasonably check both “bus” and “bike.” The event “took a bus” and the event “rode a bike” overlap, even though the survey lists them as separate response options.
The key is not whether the category labels look separate on the form. Ask whether one person can truthfully meet both definitions. A “select all that apply” format often permits overlap. A “choose exactly one” format generally makes the recorded responses disjoint, because each respondent is assigned to only one answer category. Even then, check how skipped answers and “other” responses are handled.
Being disjoint is also different from covering everyone. Categories are exhaustive for a group if every respondent belongs to at least one of them. Two categories can be disjoint but not exhaustive: for example, “uses a bus” and “rides a bike” may leave out people who walk. Conversely, two categories can cover many respondents while still overlapping.
Check the Question and the Data
A reliable check has two parts. First, read the question wording and response instructions. Does it require one answer, or allow several? Second, if respondent-level data are available, look for a person recorded in both categories. A single confirmed respondent in both is enough to show the categories are not disjoint.
Counts can give a warning, but they do not always settle the question. Suppose there are \(n\) respondents, and two category counts add to more than \(n\). If each respondent is counted no more than once in each category, then some respondents must be counted in both. But if the two counts add to \(n\) or less, that does not prove the categories are disjoint: there may still be an overlap, along with people in neither category.
When the overlap is known, the general addition rule from The General Addition Rule adjusts for the double count. When the categories are disjoint, the addition rule for mutually exclusive events from The Addition Rule for Mutually Exclusive Events applies directly. The first task is to decide which situation the survey actually describes.
Worked Example: Travel Modes on a Select-All Survey
An invented survey asks 80 students which travel modes they used to get to school during the past week. Students may select all that apply. Of the 80 respondents, 35 selected “bus,” 28 selected “bike,” and 9 selected both. Find the number and proportion who used at least one of these two modes.
State: Let \(B\) be the event that a randomly selected respondent used a bus, and let \(K\) be the event that the respondent rode a bike. The requested group is \(B\cup K\): students who used a bus, a bike, or both.
Plan: The select-all instructions allow overlap, and the data confirm that 9 students selected both. The categories are not disjoint, so adding 35 and 28 without adjustment would count those 9 students twice. Use the general addition rule, with the same 80 respondents as the denominator.
Do: First find the number who used at least one mode. Adding the two category counts counts the 9 students in both categories twice, so subtract that overlap once:
As a count-based check, the groups are 26 bus only \((35-9)\), 19 bike only \((28-9)\), and 9 both. Their total is \(26+19+9=54\). The proportion of all 80 respondents who used at least one mode is:
Conclude: In this invented survey, 54 of the 80 students, or 0.675 (67.5%), reported using a bus, a bike, or both during the week.
When Response Categories Are Disjoint
A survey’s recorded categories can be disjoint by design. Suppose a question asks, “Which one of these options was your main way of getting to school yesterday?” and instructs each respondent to choose exactly one. For that recorded response, a person cannot be counted in both “walk” and “bus.” The answer categories are disjoint, even if the person used both modes at different times in real life. The categories describe the one response recorded for the question, not every activity the person may have done.
The distinction depends on what event the problem asks about. “Selected bus as the main mode” is different from “used a bus at any time.” The first may be a single-choice category; the second may include respondents who also used other modes. State the event in words before deciding whether categories overlap.
Worked Example: One Main Study Location
An invented survey asks 150 students to choose exactly one place where they did most of their studying during the previous week. The responses are 62 for the library, 41 for home, 33 for a study room, and 14 for another place. Find the probability that a randomly selected respondent chose the library or home.
State: Let \(L\) mean “chose the library” and \(H\) mean “chose home” as the one main study location.
Plan: Each student was required to record exactly one answer. Therefore, for this survey question, a student cannot be in both \(L\) and \(H\); the recorded categories are disjoint. The counts cover all 150 respondents because the question required one of the listed answers. Add the two relevant counts and divide by 150.
Do: The number who chose either the library or home is \(62+41=103\). Thus:
The full set of response counts checks against the survey total: \(62+41+33+14=150\). The response instructions, not just that arithmetic check, establish that each student belongs to exactly one category.
Conclude: For a respondent selected at random from these 150 students, the probability of having chosen the library or home as the main study location is about 0.6867.
Use Response Patterns to See the Overlap
When individual responses are available, a small response-pattern table can make the check especially clear. For two yes-or-no categories, each respondent belongs to exactly one of four patterns: A only, both, B only, or neither. These patterns are disjoint because one respondent can have only one complete pattern. The categories “A” and “B,” however, overlap whenever the “both” count is positive.
This is the same event structure shown with Venn diagrams in Venn Diagrams for Disjoint and Overlapping Events. The table makes the respondent counts explicit: the “both” cell is the overlap, and it must be counted only once when finding how many respondents belong to either category.
Worked Example: Checking Two Newsletter Choices
An invented community survey records whether each of 72 residents reads a printed neighborhood newsletter and whether each listens to its audio version. A response-pattern tally gives the following counts:
| Listens to audio: Yes | Listens to audio: No | Total | |
|---|---|---|---|
| Reads print: Yes | 11 | 18 | 29 |
| Reads print: No | 14 | 29 | 43 |
| Total | 25 | 47 | 72 |
Determine whether reading print and listening to audio are disjoint, and find the proportion who do at least one.
State: Let \(P\) be the event that a randomly selected resident reads the print newsletter, and let \(A\) be the event that the resident listens to the audio version.
Plan: Inspect the cell where both answers are Yes. If its count is positive, at least one resident belongs to both categories, so the events are not disjoint. To find the proportion who do at least one, count the A-only group, the both group, and the P-only group once each, then divide by 72.
Do: The both-Yes cell contains 11 residents. Therefore, reading print and listening to audio are not disjoint. The number who do at least one is:
The same result follows from the category totals and overlap: the print total is 29, the audio total is 25, and the overlap is 11. Thus \(29+25-11=43\). The response-pattern cells also check: \(11+18+14+29=72\). The desired proportion is:
Conclude: The two categories overlap because 11 residents do both. In this invented survey, about 0.5972 of the residents read the print newsletter, listen to the audio version, or do both.
Common Mistakes and AP Exam Tips
- Assuming separate answer labels mean disjoint categories. A respondent may check more than one label. A full-credit explanation refers to the question’s instructions or to evidence in the data, not just the appearance of the response options.
- Adding counts from a select-all question as if each person answered once. If a respondent is in both categories, the direct sum counts that person twice. Identify the overlap and subtract it once when finding how many are in either category.
- Treating a sum no greater than the sample size as proof of no overlap. The count sum alone may not reveal whether there are people in both categories. Check the wording or examine individual responses; a positive “both” count establishes overlap.
- Confusing “disjoint” with “exhaustive.” Disjoint means no one is in both categories. Exhaustive means everyone is in at least one. State which property the question asks about.
- Mixing up the recorded answer and the real-world behavior. A choose-one question about a main preference creates one recorded category per respondent. It does not claim that the respondent has never done anything in another category.
- Using the wrong denominator. For a randomly selected respondent from the surveyed group, use the number of respondents in that group. If some people skipped the question, be clear whether the denominator is all surveyed people or only those who answered.
- Confusing overlap with independence. These are different ideas, as explained in Mutually Exclusive Versus Independent Events. For this task, the immediate check is whether any respondent belongs to both categories.
A strong AP response names the two categories as events, describes whether one respondent can meet both definitions, and uses the survey wording or the overlap data to justify the conclusion. If the events overlap, do not claim that their counts or probabilities can simply be added. If they are disjoint, say why no respondent or outcome can belong to both.
Check Your Understanding
For each situation, decide whether the categories are disjoint from the information given. Explain what evidence you used.
- A select-all survey reports 26 people choosing “walk,” 19 choosing “bike,” and 7 choosing both. Are the two categories disjoint? How many chose at least one?
- A survey requires each respondent to choose exactly one favorite fruit. Can one respondent be counted in both “apple” and “orange” for that recorded response? What fact supports your answer?
- There are 50 respondents. Two category counts are 20 and 24. Does their sum prove the categories are disjoint? Explain.
- In a 2-by-2 response-pattern table, the both-Yes cell has a count of zero. What does that tell you about the overlap in these data?
- Explain how two categories can be disjoint without being exhaustive.