When “A or B” Events Can Overlap
In The Addition Rule for Mutually Exclusive Events, you added probabilities directly when two events could not occur together. Many real situations are different: a student may take both art and music, for example. If we add the probability of taking art to the probability of taking music, students taking both courses are counted twice. The general addition rule corrects for that overlap.
In probability, “A or B” means that event A occurs, event B occurs, or both events occur. It is an inclusive “or.” The event “takes art or music” therefore includes students taking only art, only music, and both. This is the event \(A\cup B\), also called the union of A and B.
The formula can be derived by dividing the outcomes into three non-overlapping groups: those in A only, those in B only, and those in both A and B. The union contains each of these groups once. But when \(P(A)\) and \(P(B)\) are added, the outcomes in both are included once in \(P(A)\) and once in \(P(B)\). Subtracting \(P(A\cap B)\) removes one extra count, leaving the overlap included exactly once in the union.
Subtracting the overlap once from the first expression gives the second. This reasoning matches the Venn-diagram regions from Drawing a Venn Diagram of Two Events: the overlap belongs to both circles, but it is still just one part of the union.
Worked Example with Art and Music Enrollment
Suppose a school has 40 students in a particular grade. A student is selected at random from the grade. In this invented example, 18 students take art, 14 take music, and 6 take both courses. What is the probability that the selected student takes art or music?
Worked Example: Taking Art or Music
State: Let \(A\) be the event that the selected student takes art, and let \(M\) be the event that the student takes music. We want \(P(A\cup M)\), the probability that the student takes at least one of the two courses.
Plan and check: The student can take both courses, so \(A\) and \(M\) are not mutually exclusive. The group contains 40 students, and selection is at random, so each student in the grade has the same chance of being selected. Use the general addition rule, including the number taking both courses.
Do: The probabilities are \(P(A)=18/40\), \(P(M)=14/40\), and \(P(A\cap M)=6/40\). Therefore,
Check by counting students who take at least one course. There are \(18-6=12\) taking only art and \(14-6=8\) taking only music. Adding those groups and the 6 taking both gives \(12+8+6=26\) students. Thus, \(26/40=0.65\), the same result. The subtraction in the rule prevents the 6 students in both courses from being counted twice.
Conclude: The probability that a randomly selected student in this grade takes art or music is \(0.65\), or 65%. In this group, 26 of the 40 students take at least one of the two courses.
Notice that the overlap is not removed from the event. Students taking both art and music still satisfy “art or music.” The subtraction only corrects the double count created by adding the two individual probabilities.
Using “Neither” to Find the Overlap
Sometimes a problem gives the number of people in neither event instead of directly giving the overlap. The complement idea from The Probability of “Not” and “Neither” Events helps: everyone is either in the union or outside it. Once the number in the union is known, the general addition rule can be rearranged to find how many are in both.
Worked Example: Finding How Many Take Both Courses
In another invented school example, 80 students are represented. Of these, 31 take art, 28 take music, and 34 take neither course. A student is selected at random. Find the probability that the student takes art or music, and determine how many take both.
Define the events: Let \(A\) mean the student takes art and \(M\) mean the student takes music. Taking neither means the student is outside \(A\cup M\).
Find the union count: There are \(80-34=46\) students taking art, music, or both. Because the student is selected at random from the 80 students,
Find the overlap: Write the general addition rule with the counts over the same total:
The combined art and music counts total \(31+28=59\), which is 13 more than the 46 students in the union. Those 13 extra counts come from students counted in both course totals. Therefore, \(P(A\cap M)=13/80\), so 13 students take both. Check the result in the rule: \(31+28-13=46\), the union count already found.
Conclude: The probability that a randomly selected student takes art or music is \(46/80=0.575\), or 57.5%. There are 13 students who take both courses.
Applying the Rule in Another Setting
The same rule applies to any two events defined for the same chance process. The events do not need to be course choices. Be careful that each probability refers to the same group or sample space, and that the overlap describes individuals satisfying both event definitions.
Worked Example: Borrowing an E-Book or an Audiobook
Imagine a library survey of 150 patrons. In this invented example, 63 patrons borrowed an e-book during a stated period, 54 borrowed an audiobook, and 27 borrowed both. If one of the 150 patrons is selected at random, what is the probability that the patron borrowed an e-book or an audiobook?
Define and choose a rule: Let \(E\) be the event that a patron borrowed an e-book, and \(B\) the event that a patron borrowed an audiobook. A patron may have borrowed both, so use the general addition rule rather than adding the two probabilities without adjustment.
Calculate: The probabilities are \(63/150\), \(54/150\), and \(27/150\) for e-book, audiobook, and both, respectively.
For a direct check, 63 patrons borrowed an e-book; among them, 27 also borrowed an audiobook, leaving \(63-27=36\) e-book-only patrons. The audiobook-only count is \(54-27=27\). The union count is \(36+27+27=90\), so \(90/150=0.60\), agreeing with the rule.
Conclude: The probability that a randomly selected patron from this survey group borrowed an e-book or an audiobook is 0.60, or 60%. This includes patrons who borrowed both types.
Common Mistakes and AP Exam Tips
- Adding the two probabilities without checking for overlap. Direct addition is correct only when the events are mutually exclusive. If a student can take both courses, include \(P(A\cap B)\) and subtract it once.
- Subtracting the overlap twice. The union includes the shared outcomes. Subtract only one copy of the overlap because adding \(P(A)\) and \(P(B)\) counted it twice, and the union should count it once.
- Interpreting “or” as “one but not both.” Probability “or” includes the overlap. If a question specifically asks for A only or B only, that is a different event and needs a different count.
- Using counts from different groups or denominators. Make sure the art count, music count, overlap, and total describe the same set of students. If converting counts to probabilities, use the same total for all three.
- Forgetting the context in the conclusion. State what the calculated probability means and identify the group from which the individual is selected.
A clear AP-style response names the events, recognizes whether they overlap, shows the rule with the relevant values, and gives a conclusion in context. For example: “Some students take both art and music, so the events overlap. Subtracting the 6 students in both from the sum of the course counts gives \(18+14-6=26\) students in the union. Therefore, the probability that a randomly selected student takes art or music is \(26/40=0.65\).”
Check Your Understanding
For each situation, define the events, decide whether they overlap, and show how the general addition rule applies where appropriate.
- In a grade of 60 students, 24 take art, 19 take music, and 7 take both. What is the probability that a randomly selected student takes art or music?
- In a group of 90 students, 35 take art, 30 take music, and 40 take neither. How many take both, and what is the probability that a randomly selected student takes at least one course?
- A community survey includes 100 people: 42 use a bicycle for commuting, 37 use a bus, and 15 use both. Find the probability that a randomly selected person uses a bicycle or a bus.
- Explain why the probability of taking art or music includes students taking both, even though the overlap is subtracted in the formula.
- If events A and B are mutually exclusive, what is \(P(A\cap B)\), and how does the general addition rule simplify?