Question 35·Medium·Probability and Conditional Probability
A company surveyed its employees about two factors: whether they participate in the company’s wellness program and whether they usually work remotely or on-site. The results are shown below.
| Participate | Do not participate | Total | |
|---|---|---|---|
| Remote workers | 48 | 32 | 80 |
| On-site workers | 42 | 78 | 120 |
| Total | 90 | 110 | 200 |
If one employee who does not participate in the wellness program is selected at random, what is the probability that the employee is a remote worker?
For two-way table probability questions, always start by spotting any condition words like "given that," "who do not participate," or "among." That condition tells you which row or column becomes your new total, and its total gives the denominator. Then, within that restricted row or column, count how many entries also have the second characteristic (this gives the numerator). Write the fraction (numerator over denominator), then quickly simplify it; this approach avoids mixing up totals from the wrong parts of the table.
Hints
Focus on the right group
The employee is chosen from those who do not participate. Which column of the table shows the counts for non-participating employees?
Decide on the denominator
Once you focus on the "Do not participate" column, what is the total number of employees in that column? That total will be the denominator of your probability.
Find the matching numerator
Among the employees who do not participate, how many are remote workers? Put that number over the total from the previous hint to form your fraction.
Don’t forget to simplify
After you form the fraction for the probability, check if both numerator and denominator have a common factor and simplify the fraction to lowest terms.
Desmos Guide
Compute and simplify the fraction
In Desmos, type 32/110 and press Enter. Look at the simplified fraction that Desmos displays; that simplified fraction is the desired probability.
Step-by-step Explanation
Identify the given condition
The question says: "one employee who does not participate in the wellness program is selected." This means we are only interested in employees in the "Do not participate" column of the table.
So, our new total group (the denominator for the probability) is the total number of non-participating employees, which is 110.
Find how many in that group are remote
Within the "Do not participate" column, look at how many of those employees are remote workers.
From the table:
- Remote workers who do not participate: 32
This number (32) will be the numerator of our probability, because it is the count that satisfies both conditions:
- does not participate
- is a remote worker.
Write the conditional probability as a fraction
The probability we want is:
- Number who are remote and do not participate over
- Total number who do not participate
So the fraction is
This is the probability before simplifying.
Simplify the fraction to get the final answer
Now simplify .
Both 32 and 110 are divisible by 2:
So the probability that a randomly selected non-participating employee is a remote worker is , which corresponds to choice B.