Question 947·Medium·Math

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At a community center, 180 students are enrolled in at least one after-school activity. Of these students, 75 are enrolled in the robotics club. Of the students enrolled in robotics, $\frac{2}{5}$ attend an activity session on Monday. Of the students not enrolled in robotics, $\frac{1}{3}$ attend an activity session on Monday.

If one student who attends an activity session on Monday is selected at random, what is the probability that the student is enrolled in the robotics club?

Your answer

A conditional probability question limits who can be picked before asking what else is true about that person. If the selection is from Monday attendees, count attendees in each category and add them for the denominator. Use the attendees in the target category for the numerator. Evaluate each count in Desmos; using the original total would include people who cannot be picked.

Hints

Hint 1
The fractions describe Monday attendance within each robotics group, not robotics membership among Monday attendees. Multiply each fraction by the size of its own group to find how many from that group attend on Monday.
Hint 2
Students not in robotics are everyone left after removing robotics students from the full group. Find that count before using the one-third Monday attendance rate.
Hint 3
The student is selected from Monday attendees, so add the Monday counts from both groups to get the denominator. Which of those attendees belong in the numerator?

Step-by-step

Count both Monday groups
Step 1
Count robotics students who attend Monday
The problem gives 75 robotics students, and $\frac{2}{5}$ of them attend Monday. Type $r=75\cdot\frac{2}{5}$ in Desmos; it shows $r=30$ . Let $r$ mean robotics students who attend Monday. These 30 are in both groups the question names.
Step 2
Find the size of the other group
The 180 students include the 75 in robotics. Type $n=180-75$ in Desmos; it shows $n=105$ , the number not in robotics.
Step 3
Count other students who attend Monday
Of the 105 students not in robotics, $\frac{1}{3}$ attend Monday. Type $m=n\cdot\frac{1}{3}$ in Desmos; it shows $m=35$ . Let $m$ mean Monday attendees not in robotics.
Step 4
Build the group the student is picked from
The condition, what you already know about the selected student, is that the student attends Monday. So the possible picks are all Monday attendees: robotics attendees plus non-robotics attendees. Type $t=r+m$ in Desmos; it shows $t=65$ . Use this Monday total, not all 180 students, as the denominator.
Step 5
Find the probability within Monday attendees
Of the 65 possible picks, 30 are in robotics. Type $r/t$ in Desmos; it shows $0.461538\ldots$ . Tap the fraction button to see $\frac{6}{13}$ . So the probability that a selected Monday attendee is in robotics is $\frac{6}{13}$ . Grid in 6/13.

Lessons that teach this

SAT Data AnalysisIntermediateCoreUse two-way tables and conditional probability

Question 947·Medium·Math

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