Question 14·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

0 / 50 Questions completed

At a high school, some students participate in robotics, debate, or both activities. Of the students who participate in robotics, $40\%$ also participate in debate. Of the students who participate in debate, $25\%$ also participate in robotics.

A student is selected at random from the students who participate in exactly one of the two activities. What is the probability that the selected student participates in robotics?

Your answer

The phrase “selected from exactly one activity” signals conditional probability: only students in that group belong in the denominator. Use the students who do both activities as a common anchor, then find each activity’s total with Desmos regressions. Subtract the overlap before finding the probability. Don’t combine the leftover percentages directly, because they’re percentages of different-sized groups.

Hints

Hint 1
The overlap means students who do both activities. Both percentages count these same students, but each uses a different activity group as its whole. How could you give the overlap a convenient count?
Hint 2
No total number of students is given, so try 10 students in both activities. Use $40\%$ to find the robotics total and $25\%$ to find the debate total. Scaling every count would leave the probability unchanged.
Hint 3
“Exactly one” keeps robotics only and debate only. Subtract the shared students from each activity total. Which of those two remaining counts belongs in the numerator?

Step-by-step

Use a shared count to build the eligible group
Step 1
Give the overlap a common count
The overlap is the students who do both activities. The two percentages describe this same group, even though they’re taken from different totals. Call the robotics total $R$ and the debate total $D$ . Choose $10$ students in the overlap as a convenient model. Then the given statements become $0.40R=10\qquad\text{and}\qquad0.25D=10$ . The school’s actual counts could be scaled up or down together; that scaling won’t change the probability.
Step 2
Find the robotics total
Type $40\%\operatorname{of}r_{1}\sim10$ , where $r_{1}$ is the robotics total. The $\sim$ asks Desmos to find a value that fits the equation. Under PARAMETERS, Desmos shows $r_{1}=25$ , so the model has $25$ robotics participants.
Step 3
Find the debate total
Keep the first line and type $25\%\operatorname{of}d_{1}\sim10$ , where $d_{1}$ is the debate total. Under PARAMETERS, Desmos shows $d_{1}=40$ , so the model has $40$ debate participants.
Step 4
Count students in exactly one activity
Count the overlap in each activity total, but exclude it from the exactly-one group. Subtract the $10$ shared students from each total: $\text{robotics only}=25-10=15$ $\text{debate only}=40-10=30$ These are the two groups eligible to be selected.
Step 5
Find the probability within the eligible group
The numerator is the $15$ robotics-only students; the denominator includes everyone in exactly one activity. Type $\frac{15}{15+30}$ in Desmos. It displays $0.333\ldots$ ; its fraction button gives $\frac{1}{3}$ . So the probability that the selected student participates in robotics is $\frac{1}{3}$ . Grid in 1/3 (or .3333).

Question 14·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this