Super-Hard SAT Data Analysis Question 6 | Free SAT Prep

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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?

For a conditional-probability question involving several groups, first turn every given proportion into a count. Then restrict the denominator to the group named after “given” or described by the condition. Finally, count only the favorable members within that restricted group. Keeping the groups separate until the final fraction helps prevent mixing an overall probability with a conditional probability.

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Separate the groups

There are three nonoverlapping groups: weekday only, weekend only, and both weekday and weekend shifts.

Use the condition in the question

Because the selected volunteer has completed training, the denominator should be the total number of trained volunteers, not $480$ .

Identify favorable trained volunteers

A trained volunteer has at least one weekend shift if the volunteer is in either the weekend-only group or the both-shifts group.

Desmos Guide

Calculate the trained counts

Direct arithmetic is likely faster, but Desmos can verify the counts. Enter these expressions on separate lines:

480(0.35)(5/7)

480(0.40)(3/8)

480(0.25)(3/4)

Calculate the conditional probability

Enter

(480(0.40)(3/8)+480(0.25)(3/4))/(480(0.35)(5/7)+480(0.40)(3/8)+480(0.25)(3/4))

Then convert the displayed result to a fraction or compare its decimal value with the answer choices.

Step-by-step Explanation

Find the number of volunteers in each group

The numbers scheduled for weekday only, weekend only, and both types of shifts are

\begin{aligned} 0.35(480) &= 168 \\ 0.40(480) &= 192 \\ 480-168-192 &= 120 \end{aligned}

Find the trained volunteers in each group

Multiply each group total by its training fraction.

\begin{aligned} \text{Weekday only: }&168\left(\frac{5}{7}\right)=120 \\ \text{Weekend only: }&192\left(\frac{3}{8}\right)=72 \\ \text{Both shifts: }&120\left(\frac{3}{4}\right)=90 \end{aligned}

Thus, the total number of trained volunteers is $120+72+90=282$ .

Restrict to trained volunteers with a weekend shift

Among trained volunteers, those scheduled for at least one weekend shift are the $72$ weekend-only volunteers and the $90$ both-shifts volunteers. Therefore, the probability is

\frac{72+90}{282}=\frac{162}{282}=\frac{27}{47}.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation