Question 995·Hard·Math

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Work status	Engineering	Marketing	Finance
Remote	21	18	x
On-site	35	25	17

The table above shows the numbers of employees at a company, classified by department and by whether they work remotely or on-site.

If an employee is selected at random:

The probability that the employee belongs to the Engineering department given that the employee works remotely is $\dfrac{3}{7}$ .
The probability that the employee works remotely is $\dfrac{7}{18}$ .

What is the value of $x$ ?

Your answer

A “given that” statement in a two-way table signals a conditional probability: restrict to the given group before choosing the denominator. Put the target cell over the named row or column total, then use a Desmos regression to find the missing count. Don’t include employees outside that group. A probability without “given that” instead uses the whole company.

Hints

Hint 1
In a conditional probability, “given that” limits who can be selected. Since the employee works remotely, use only the remote row for the denominator, even though the table also lists on-site engineers.
Hint 2
The numerator is the remote Engineering count. Add the three cells in the remote row for the denominator. What equation makes that fraction equal the given probability?
Hint 3
A Desmos regression can find the missing count from your equation. Change $x$ to $x_1$ and $=$ to $\sim$ so Desmos solves for the unknown cell.

Step-by-step

Approach 1: Use the remote-only probability

Step 1
Find the remote row total
Call the total number of remote employees $R$ . The remote row has $21$ in Engineering, $18$ in Marketing, and $x$ in Finance, so $R=21+18+x$ . The on-site counts don't belong in this remote row total.
Step 2
Write the conditional probability equation
A conditional probability is a chance measured within a group you already know the employee belongs to. Given that the employee works remotely makes the remote row the denominator; its $21$ engineers are the numerator. So $\frac{21}{21+18+x}=\frac{3}{7}$ . Using the whole company as the denominator would answer a different question.
Step 3
Solve for the missing Finance count
Type $\frac{21}{21+18+x_1}\sim\frac{3}{7}$ in Desmos. The $\sim$ tells Desmos to fit the unknown $x_1$ instead of graphing an equation in $x$ . Under PARAMETERS, it shows $x_1=10$ . So 10 employees work remotely in Finance. Grid in 10.

Approach 2: Use the company-wide probability

Step 1
Count the on-site employees
The on-site row contains $35$ , $25$ , and $17$ employees. Type $35+25+17$ in Desmos; it shows $77$ . That's the on-site total.
Step 2
Write a probability using the whole company
The probability of working remotely has no “given that” condition, so its denominator is all employees. Let $n$ be the remote total. With $77$ on-site employees, the company total is $n+77$ , so $\frac{n}{n+77}=\frac{7}{18}$ .
Step 3
Find the remote total
Add $\frac{n}{n+77}\sim\frac{7}{18}$ in Desmos. Under PARAMETERS, it shows $n=49$ , the number of remote employees across all departments.
Step 4
Remove the two known remote counts
The $49$ remote employees include $21$ in Engineering and $18$ in Marketing. Type $n-21-18$ in Desmos; it shows $10$ . The employees left are the remote Finance employees. Grid in 10.

Lessons that teach this

SAT Data AnalysisIntermediateCoreUse two-way tables and conditional probability

Question 995·Hard·Math

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Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

Lessons that teach this