Question 20·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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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?

A conditional probability starts by narrowing the group to students who meet every given condition. For “one or both” services, add the two service counts and subtract those counted twice. Do that for all visitors and for workshop attendees, then remove the workshop attendees. Count the students who used both services within the same narrowed group before dividing.

Hints

Hint 1
The word given restricts the sample space, the students who could be selected. Here, a student must be outside the workshop and have used at least one service. How can you count everyone who used at least one?
Hint 2
The union of two service groups means everyone who used one or both. Add the group counts, then subtract those in both groups once because you counted them twice. Make this count for all visitors and for workshop attendees.
Hint 3
To find the eligible group, remove workshop service users from all service users. Then remove workshop students who used both services from the total who used both. Which count belongs on the bottom of the probability?

Step-by-step

Count only eligible students
Step 1
Count everyone who used at least one service
Type $120+96-48$ in Desmos; it shows $168$ . The union is everyone who used one or both services. Adding $120$ and $96$ counts the $48$ who used both twice, so subtract them once.
Step 2
Count workshop attendees who used a service
Make the same count inside the workshop. Type $18+14-8$ on the next line; Desmos shows $24$ . The $8$ who used both services were counted twice in the addition, so subtract them once.
Step 3
Find the group the student can come from
The student did not attend the workshop, so remove those $24$ workshop service users from the $168$ service users. Type $168-24$ ; Desmos shows $144$ . That is the denominator, not all $240$ visitors or all $168$ service users.
Step 4
Count eligible students who used both services
The $48$ students who used both services include $8$ workshop attendees. Type $48-8$ ; Desmos shows $40$ students who used both outside the workshop. Keep the numerator and denominator inside the same conditioned group.
Step 5
Find the conditional probability
A conditional probability divides the favorable count by the eligible count. Type $\frac{40}{144}$ ; Desmos shows $0.2777\ldots$ . Tap the fraction button to see $\frac{5}{18}$ . So the probability that an eligible student used both services is $\frac{5}{18}$ . Choice C.

Lessons that teach this

SAT Data AnalysisIntermediateCoreUse two-way tables and conditional probability

Question 20·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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