Turn a Word Problem into an Outcome Check
In What Mutually Exclusive Events Mean, you learned that two events are mutually exclusive, or disjoint, when they have no outcomes in common. This tutorial practices applying that definition to word problems, where the most important step is often figuring out exactly what one outcome represents.
A word problem may describe people, objects, days, or stages of a process. Before deciding whether two events are disjoint, pin down the chance process: What is selected, observed, or recorded? What information is included in one complete outcome? Then ask whether one such outcome could make both event descriptions true.
This method focuses on possibility, not on whether the overlap is common. Even one shared outcome is enough to show that events are not disjoint. Conversely, events are disjoint only when the chance process and the event definitions rule out every shared outcome.
Pay particular attention to words that describe how information is recorded. “Recorded as exactly one category” may rule out overlap. “Has,” “uses,” or “includes” often describes a characteristic that can coexist with another characteristic. The labels alone do not settle the question; the outcome and the event definitions do.
A Practical Checklist for Word Problems
Identify what happens once: one item is selected, one day is observed, or one complete set of stages is recorded.
For a single selection, an outcome might be one particular book. For a two-stage process, it might be an ordered pair recording both stages.
Write what must be true for an outcome to belong to each event.
Try to name or construct one outcome that satisfies both descriptions. A single valid example establishes overlap.
Explain whether the events are mutually exclusive and give the outcome-based reason.
When a problem gives a list, a table, or a set of counts, use it to make the check concrete. Look for an individual item in both groups, or use the stated classification rules to determine whether that is possible. Do not infer that groups are disjoint just because their totals are reported separately.
The event notation from earlier in the course is useful shorthand: \(A\cap B\) represents the event that both \(A\) and \(B\) occur. Disjointness means \(A\cap B=\varnothing\). For this tutorial, the key task is identifying whether that intersection contains any possible outcomes—not calculating a probability.
Worked Example: One Primary Service at a Repair Desk
A repair desk records exactly one primary service for each visit: screen repair, battery replacement, or software help. An invented log contains 28 screen-repair visits, 19 battery-replacement visits, and 13 software-help visits, for 60 visits in total. One visit is selected at random. Let \(S\) be the event that the visit is recorded as screen repair, and let \(B\) be the event that it is recorded as battery replacement. Are \(S\) and \(B\) mutually exclusive?
Define one outcome: One outcome is the primary-service category recorded for one visit. The problem says each visit receives exactly one primary-service category.
Check for a shared outcome: For a visit to belong to \(S\), its recorded category must be screen repair. For it to belong to \(B\), its recorded category must be battery replacement. Because one visit cannot receive both primary categories under the stated rule, no possible outcome meets both descriptions. Thus, \(S\cap B=\varnothing\).
Conclude: The events are mutually exclusive in this recording system. The justification is the rule that every visit has exactly one primary-service category. The counts, 28 and 19, are consistent with that rule, but the classification follows from the rule rather than from the sizes of the counts.
If the question instead asked whether a visit received screen repair and also received a battery replacement, the outcome would need to record all services performed. A visit could then meet both descriptions. The word “primary” and the exactly-one rule are what make the events disjoint here.
Different Characteristics Can Overlap
Events often describe different characteristics of the same object. A book can be both a mystery and a paperback; a person can use a bicycle and wear a helmet. When a word problem describes characteristics rather than one-choice categories, actively look for an item that has both.
Worked Example: A Novel That Is Both a Mystery and a Paperback
A library inventory lists 84 books. Among them, 31 are mysteries, 46 are paperbacks, and 18 are both mysteries and paperbacks. One book is selected at random. Let \(M\) be the event that the selected book is a mystery and \(P\) the event that it is a paperback. Are \(M\) and \(P\) mutually exclusive?
Define one outcome: One outcome is the identity of the selected book. Its outcome includes all of its characteristics, such as its genre and format.
Check for a shared outcome: The inventory reports 18 books that are both mysteries and paperbacks. Each of those books satisfies the condition for \(M\) and the condition for \(P\). Therefore, \(M\cap P\) contains 18 possible book outcomes and is not empty.
Conclude: The events are not mutually exclusive. For example, any one of the 18 books in both groups is a shared outcome. It does not matter that the event descriptions name different characteristics; one book can have both.
A frequent reasoning trap is to say that the events are disjoint because “mystery” is a genre and “paperback” is a format. Those are different kinds of labels, but they can describe the same book. The count of 18 directly confirms that overlap.
For Multi-Stage Processes, Keep the Whole Outcome
A complete outcome must include every stage named in the chance process. This point matters when one event describes the first stage and another describes a feature of the whole result. If you record only part of the process, you may miss whether an outcome satisfies both event descriptions.
For example, if two number cubes are rolled together, a complete outcome is an ordered pair \((\text{first result},\text{second result})\). “The first cube shows 4” and “the sum is 7” can both be true for the outcome \((4,3)\). Checking only the first cube’s result or only the sum would not show the complete shared outcome as clearly.
Worked Example: Two Number Cubes
Two fair six-sided number cubes are rolled, and the outcome records the first cube’s result followed by the second cube’s result. Let \(F\) be the event that the first cube shows 4. Let \(T\) be the event that the sum of the two results is 7. Are \(F\) and \(T\) mutually exclusive?
Define one outcome: One outcome is an ordered pair \((x,y)\), where \(x\) is the first cube’s result and \(y\) is the second cube’s result. There are \(6\times6=36\) possible ordered pairs.
Translate the events: An outcome belongs to \(F\) when its first coordinate is 4. It belongs to \(T\) when its two coordinates add to 7.
Search for a shared outcome: The ordered pair \((4,3)\) has first coordinate 4, so it is in \(F\). Its sum is \(4+3=7\), so it is also in \(T\). Thus, \((4,3)\in F\cap T\), and the intersection is not empty.
Conclude: The events are not mutually exclusive. The single possible outcome \((4,3)\) satisfies both event descriptions. In fact, \((4,3)\) is the only shared outcome, because when the first result is 4, the second must be 3 to make a sum of 7. One shared outcome is enough to establish that the events overlap.
Notice that this reasoning uses the complete ordered pair. If the outcome were recorded only as the sum, the event “the first cube shows 4” could not be determined from that reduced record. Always use an outcome definition that preserves the information required by the events.
When the Wording Does Not Settle the Classification
Sometimes a word problem leaves the recording rule unclear. In that case, do not make up a restriction that the question has not stated. Identify the ambiguity and explain how different reasonable interpretations affect the classification.
Suppose a survey asks whether a visitor arrived by bus or by bicycle. If the response requires exactly one main arrival method, those recorded categories are disjoint. If visitors can select every method they used, someone who rode a bus and then a bicycle could belong to both groups. The context needs to specify what one response records before the classification can be certain.
Similarly, phrases such as “has a membership,” “used a service,” or “visited a place” do not usually imply that only one such characteristic can apply. By contrast, a stated rule such as “each participant is assigned to exactly one group” does rule out membership in two assigned groups for that classification.
Common Mistakes and AP Exam Tips
- Using the labels instead of the outcomes. Different-sounding labels can apply to the same item. A book can be a mystery and a paperback. Name a possible outcome and test both conditions on it.
- Assuming separate counts mean separate groups. A report may give one count for each characteristic even when individuals belong to both groups. Look for an overlap count, a shared example, or an explicit one-category rule.
- Checking only part of a multi-stage result. If an outcome records two stages, write both parts. A shared outcome may be visible only when the complete pair or sequence is considered.
- Confusing “rare” with “impossible.” An overlap does not disappear because it is unusual or has a small count. One possible shared outcome means the events are not mutually exclusive.
- Ignoring ambiguity in the wording. Do not assume that a person can choose only one option unless the problem says so. A complete answer identifies the missing rule and explains its effect.
- Giving only a yes-or-no answer. An AP-quality justification says what one outcome represents, identifies a shared outcome or explains why none can exist, and states the classification in context.
A full-credit response for the number-cube example could say: “A complete outcome is an ordered pair of cube results. The outcome \((4,3)\) has a first result of 4 and a sum of 7, so it belongs to both events. Therefore, the events are not mutually exclusive.” This answer supplies evidence, not just a label.
Check Your Understanding
For each situation, decide whether the two events are mutually exclusive. Explain your decision using the outcome or recording rule.
- A randomly selected calendar date is classified as either a weekday or a weekend day, with every date assigned to exactly one of those categories. Let \(W\) be “weekday” and \(E\) be “weekend day.” Are \(W\) and \(E\) disjoint? State the rule that supports your decision.
- A randomly selected student is asked whether they play soccer and whether they play chess. A student may answer yes to both. Let \(S\) be “plays soccer” and \(C\) be “plays chess.” What would establish that these events are not mutually exclusive?
- A spinner is spun twice. A complete outcome is an ordered pair of results from 1 through 4. Let \(A\) be “the first spin is 2” and \(B\) be “the two results are equal.” Give a shared outcome if one exists, and classify the events.
- A package is recorded as exactly one of three shipping methods: ground, air, or express. For one selected package, let \(G\) be “recorded as ground” and \(A\) be “recorded as air.” Are these events disjoint? Explain why the recording rule matters.
- A community survey asks whether residents own a garden and whether they own a pet, but does not report whether the groups overlap. Can you classify the two events as mutually exclusive from the separate group counts alone? Explain what additional information would help.