Question 6·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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A volunteer program has 480 volunteers. Of these volunteers, 35% are scheduled for weekday shifts only, 40% are scheduled for weekend shifts only, and the remaining volunteers are scheduled for both weekday and weekend shifts.

Of the volunteers scheduled for weekday shifts only, $\frac{5}{7}$ have completed training. Of the volunteers scheduled for weekend shifts only, $\frac{3}{8}$ have completed training. Of the volunteers scheduled for both weekday and weekend shifts, $\frac{3}{4}$ have completed training.

A volunteer who has completed training is selected at random. Which choice is the probability that the selected volunteer is scheduled for at least one weekend shift?

A conditional probability is a probability among people who meet a stated condition. The words “selected from” tell you who can be picked, so find the trained count in each shift group with Desmos. Then count which trained people have at least one weekend shift. Keep the denominator inside the selected group, not the full program.

Hints

Hint 1
The remaining shift group is a complement: it gets the percent left after the weekday-only and weekend-only shares are taken from $100\%$ . What percent of volunteers work both shifts?
Hint 2
Each training fraction uses its own shift group as the whole. Find the three shift-group counts first, then take the stated fraction of each count. None of those fractions is taken directly out of all $480$ volunteers.
Hint 3
The condition limits who could be selected: only trained volunteers qualify. Add all three trained counts for the denominator. Which trained groups meet the target of at least one weekend shift?

Step-by-step

Count trained volunteers by shift group
Step 1
Find the share working both shifts
The weekday-only and weekend-only groups don't overlap, so subtract both shares from $100\%$ . Type $100-35-40$ in Desmos; it shows $25$ . The remaining volunteers, who work both shifts, make up $25\%$ .
Step 2
Find each shift-group count
Each shift percent is taken of all $480$ volunteers. Type the list $a=[35\%\operatorname{of}480,40\%\operatorname{of}480,25\%\operatorname{of}480]$ . Desmos shows $a=[168,192,120]$ : weekday only, weekend only, then both shifts, in that order.
Step 3
Find the trained counts
Each training fraction applies to its own shift group, not all $480$ volunteers. Type $t=[\frac{5}{7}a[1],\frac{3}{8}a[2],\frac{3}{4}a[3]]$ ; $a[1]$ means the first count in $a$ . Desmos shows $t=[120,72,90]$ trained volunteers in the same group order.
Step 4
Count everyone who could be selected
The volunteer is selected from those who completed training, so all three trained groups belong in the denominator. Type $\operatorname{total}(t)$ ; Desmos shows $282$ trained volunteers.
Step 5
Count the trained volunteers with weekend shifts
At least one weekend shift includes weekend only and both shifts. Those are the second and third entries of $t$ . Type $t[2]+t[3]$ ; Desmos shows $162$ trained volunteers who meet the target.
Step 6
Divide within the trained group
The probability is the $162$ trained volunteers with weekend shifts out of all $282$ trained volunteers. Type $\frac{162}{282}$ ; Desmos shows a decimal. Click its fraction button to see $\frac{27}{47}$ . So the probability for the selected volunteer is $\frac{27}{47}$ . Choice D.

Question 6·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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