Question 35·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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At a technology conference, each of 420 attendees attended exactly one workshop. Each workshop was in either the design track or the data science track and was held either in the morning or in the afternoon. Of the attendees, 252 selected the design track.

Given that a randomly selected attendee was in the data science track, the probability that the attendee attended an afternoon workshop was $\frac{5}{8}$ . Given that a randomly selected attendee attended a morning workshop, the probability that the attendee selected the design track was $\frac{7}{10}$ .

What is the number of attendees who selected the design track and attended an afternoon workshop?

Your answer

Think of track and time as a two-way table: each attendee belongs to one track and one time of day. A probability stated “given that” applies only within its named group. Find the data science morning count, use the morning design-to-data science ratio, then subtract from the design total. Using all attendees as the base for either probability gives the wrong group.

Hints

Hint 1
Every attendee chose one of the two tracks. Subtract the design total from the overall total to find how many chose data science.
Hint 2
Within data science, morning and afternoon are the only possibilities. The complement of the afternoon probability is the morning probability. Apply it to the data science total.
Hint 3
Among morning attendees, $7$ of every $10$ chose design, leaving $3$ of every $10$ in data science. What ratio do those two morning counts have?

Step-by-step

Find the morning counts, then subtract
Step 1
Find the data science total
Each attendee chose exactly one track, so everyone not in design is in data science. Subtract the design count from the total:
$420-252=168.$
Step 2
Find the morning share of data science
Morning and afternoon use up the data science group. The morning share is the complement, or the part left after the afternoon share. Subtract $\frac{5}{8}$ from $1$ :
$1-\frac{5}{8}=\frac{3}{8}.$
Step 3
Count data science attendees in the morning
Type $168\cdot\frac{3}{8}$ in Desmos. It shows $63$ . You’ve applied the morning share to only the data science attendees, so $63$ of them attended in the morning.
Step 4
Relate the two morning counts
The $\frac{7}{10}$ is a conditional probability: it splits the morning group, not all $420$ attendees. For every $10$ morning attendees, $7$ chose design and the other $3$ chose data science. Because both counts come from the same morning group, their ratio is $7:3$ . Let $x$ be the morning design count. Using the $63$ morning data science attendees, write:
$\frac{x}{63}=\frac{7}{3}.$
Step 5
Solve for morning design
Type $\frac{x_1}{63}\sim\frac{7}{3}$ in Desmos. Use $x_1$ for the unknown $x$ and $\sim$ to have Desmos solve the equation. Under PARAMETERS, it shows $x_1=147$ , the morning design count.
Step 6
Find afternoon design
Every design attendee attended either morning or afternoon, so subtract the morning count from all $252$ design attendees. Type $252-x_1$ on the next Desmos line. It shows $105$ , since $252-147=105$ . There were $105$ design-track attendees in afternoon workshops. Grid in 105.

Lessons that teach this

SAT Data AnalysisIntermediateCoreUse two-way tables and conditional probability

Question 35·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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