Question 914·Hard·Math

0 / 2000

At a learning center, 240 students may enroll in the Design workshop, the Robotics workshop, both workshops, or neither workshop.

Workshop enrollment	Number of students
Enrolled in Design	150
Enrolled in Robotics	130
Total students	240

If a student is selected at random from the students enrolled in exactly one of the two workshops, the probability that the student is enrolled in Design is $\frac{4}{7}$ . What is the probability that a student selected at random from the students not enrolled in Robotics is enrolled in Design?

Your answer

Overlapping workshop totals and a selection made from a smaller group signal conditional probability: count successes only inside the group that can be picked. Name the overlap so each workshop-only count is its total minus that overlap. Turn the given probability into a ratio, then use a Desmos regression to find the overlap. When the selection group changes, its denominator changes too.

Hints

Hint 1
Both workshops are included in each workshop total. If $x$ students attend both, subtract $x$ from each total to find the Design-only and Robotics-only counts.
Hint 2
The exactly-one students split into Design only and Robotics only. A probability of $\frac{4}{7}$ means that, for every $4$ Design-only students, there are $3$ Robotics-only students. What equation relates those counts?
Hint 3
Not enrolled in Robotics names a different selection group from exactly one workshop. Find its size by subtracting the Robotics total from the center total. Which students in that group also attend Design?

Step-by-step

Find the overlap, then change the denominator
Step 1
Count the students in only one workshop
Let $x$ be the number enrolled in both workshops. Those students appear in each workshop total, so the Design-only count is $150-x$ and the Robotics-only count is $130-x$ .
Step 2
Turn the given probability into a ratio
The exactly-one group contains only Design-only and Robotics-only students. The probability $\frac{4}{7}$ means $4$ of every $7$ equal parts are Design-only, leaving $3$ parts Robotics-only. Their ratio, which compares the two counts, gives the equation $\frac{150-x}{130-x}=\frac{4}{3}$ .
Step 3
Clear the fractions
Multiply both sides by $3(130-x)$ so the two workshop-only counts can be compared without fractions: $3(150-x)=4(130-x)$ .
Step 4
Find the overlap in Desmos
Replace $x$ with $x_1$ and $=$ with $\sim$ , so Desmos fits the unknown count rather than graphing it. Type $3(150-x_1)\sim4(130-x_1)$ . Under PARAMETERS, Desmos shows $x_1=70$ , so 70 students attend both workshops.
Step 5
Find the favorable count
Subtract the overlap from the Design total: $150-70=80$ . These 80 Design-only students are enrolled in Design but not Robotics.
Step 6
Find the new selection group
The complement of Robotics means everyone outside Robotics. Subtract its total from all students: $240-130=110$ . These 110 students not in Robotics include Design-only students and students in neither workshop. The exactly-one group isn't the denominator here because it includes Robotics-only students.
Step 7
Find the requested probability
Of the $110$ students not enrolled in Robotics, $80$ are enrolled in Design. So the requested probability is $\frac{80}{110}=\frac{8}{11}$ . Grid in 8/11.

Question 914·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this