In the Remote row, how many employees are in Engineering, and how many employees in total are remote? Use these two numbers to write $P(\text{Engineering} \mid \text{Remote})$ as a fraction in terms of $x$.

Set your fraction for $P(\text{Engineering} \mid \text{Remote})$ equal to $3/7$ and solve for $x$ by cross-multiplying. After you find $x$, you can check it using the overall remote probability $P(\text{Remote}) = 7/18$.

In Desmos, enter y = 21/(39 + x) on one line and y = 3/7 on another line. These represent the conditional probability from the table and the given value, respectively.

Use the graph to locate the point where the two curves intersect. Tap or click on the intersection point and read off its x-coordinate; that x-value is the one that makes the two probabilities equal and is the required value in the table.

The phrase "the employee belongs to the Engineering department given that the employee works remotely" means we only look at the Remote row.

• Number of remote Engineering employees: $21$
• Total number of remote employees: $21 + 18 + x = 39 + x$

So the conditional probability is:

We are told this equals $\dfrac{3}{7}$.

Set the expression from the table equal to the given probability:

Cross-multiply:

Simplify each side:

Now you can solve for $x$ from $30 = 3x$.

From $30 = 3x$, divide both sides by $3$:

Now check this with the second piece of information, $P(\text{Remote}) = \dfrac{7}{18}$.

• Total remote employees: $21 + 18 + 10 = 49$
• Total employees: $21 + 35 + 18 + 25 + 10 + 17 = 126$

So

which matches the given probability. Therefore, the value of $x$ is $10$.