Question 911·Medium·Math

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During a particular week, a community center assigns each of its 240 volunteers to exactly one of two roles: organizing supplies or greeting visitors. Of the volunteers, 90 organize supplies. Of the volunteers who organize supplies, $\frac{2}{5}$ work an evening shift. Of the volunteers who greet visitors, $\frac{1}{3}$ work an evening shift.

If a volunteer is selected at random from the volunteers who work an evening shift that week, which choice is the probability that the selected volunteer organizes supplies?

A conditional probability limits the pick to a stated group, signaled here by “selected from.” That group supplies the denominator, even if the problem gives a larger total. Use Desmos to count the evening workers in each role, then divide the evening organizers by all evening workers. A fraction describing organizers who work evenings answers the question in the reverse direction.

Hints

Hint 1
The two roles include every volunteer, with no overlap. Subtract the number of organizers from the total to find the greeters. That gives you the group to which the greeters’ evening-shift fraction applies.
Hint 2
Each evening-shift fraction describes one role, not all volunteers. Multiply the number of organizers by their fraction and the number of greeters by theirs. You need both evening counts before you can find the group being picked from.
Hint 3
“Selected from” tells you the condition: only evening workers can be picked. Add the evening workers from both roles for the denominator. Which part of that group organizes supplies?

Step-by-step

Count the evening workers in each role
Step 1
Find how many volunteers greet visitors
Every volunteer has exactly one role, so anyone who doesn't organize supplies greets visitors. Type $240-90$ in Desmos; it shows $150$ greeters.
Step 2
Count evening organizers
The $\frac{2}{5}$ applies only to the 90 organizers. Type $90\cdot\frac{2}{5}$ ; Desmos shows $36$ organizers working evenings.
Step 3
Count evening greeters
The $\frac{1}{3}$ applies only to the 150 greeters found in the first step. Type $150\cdot\frac{1}{3}$ ; Desmos shows $50$ greeters working evenings.
Step 4
Build the group being picked from
The denominator counts everyone who could be picked. The pick is from evening workers, including both roles, so type $36+50$ . Desmos shows $86$ evening workers. The group being picked from sets the denominator. Using $240$ would include volunteers who don't work evenings.
Step 5
Find the probability
The numerator counts the people in that group who meet the target: the $36$ evening organizers. Type $\frac{36}{86}$ ; Desmos shows about $0.4186$ , and its fraction button gives $\frac{18}{43}$ . So the probability that a selected evening worker organizes supplies is $\frac{18}{43}$ . Choice A.

Question 911·Medium·Math

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