During one week, 240 students visited a college media center. Of these students, 120 borrowed at least one book, 96 used a computer workstation, and 48 both borrowed at least one book and used a computer workstation. In addition, 40 students attended a research workshop. Among the students who attended the workshop, 18 borrowed at least one book, 14 used a computer workstation, and 8 both borrowed at least one book and used a computer workstation.
A student is selected at random from the 240 students. Given that the student did not attend the research workshop and used one or both of the following services—borrowing at least one book and using a computer workstation—which choice is the probability that the student both borrowed at least one book and used a computer workstation?
For a conditional-probability question involving overlapping groups, first identify the restricted group named after “given that”; that group becomes the denominator. Use addition and subtraction to avoid double-counting students in both categories. Then count only the favorable students who also satisfy every part of the condition before forming and simplifying the fraction.
Hints
Use inclusion-exclusion
To count students who borrowed a book or used a workstation, add the two service totals and subtract the students who used both services.
Apply the workshop restriction
Find how many workshop attendees used at least one of the two services. Subtract that count from the overall count of students who used at least one service.
Build the conditional fraction
For the numerator, start with the students who used both services and remove the workshop attendees from that group. Divide by the restricted group from the previous hint.
Desmos Guide
Calculate the restricted total
Algebra is fastest for this problem. In the Desmos calculator, enter 120+96-48 to find the number who used at least one service. Then enter (120+96-48)-(18+14-8) to remove workshop attendees who used at least one service.
Calculate the conditional probability
Enter (48-8)/((120+96-48)-(18+14-8)). Desmos will simplify the fraction, which can be matched to the answer choices.
Step-by-step Explanation
Count students who used at least one service
Add the numbers who borrowed a book and who used a workstation, then subtract the 48 students counted twice:
students borrowed a book or used a workstation.
Remove workshop attendees from the condition group
Among workshop attendees, the number who borrowed a book or used a workstation is .
Therefore, the number of students who did not attend the workshop and used at least one service is .
Count favorable students and form the probability
Of the 48 students who both borrowed a book and used a workstation, 8 attended the workshop. Thus, favorable students did not attend the workshop.
The conditional probability is .