Question 42·Hard·Probability and Conditional Probability
The table below summarizes enrollment information for all 12,000 undergraduate students at a university this semester.
| Enrolled in a language course | Not enrolled in a language course | |
|---|---|---|
| Enrolled in a science course | 2,700 | 1,800 |
| Not enrolled in a science course | 3,900 | 3,600 |
To the nearest tenth of a percent, what percent of students enrolled in a language course are also enrolled in a science course?
For two-way table questions asking for a percent or probability "of students who ___, how many also ___", always start by identifying the conditioning group from the wording (the part after "of students who..."). Use the total of that row or column as your denominator, then use the single cell where both conditions are true (the intersection of the row and column) as your numerator. Form the fraction numerator/denominator, convert it to a decimal if needed, then to a percent, and finally round as the question specifies—ignore the overall total unless the question explicitly asks about all students.
Hints
Identify the reference group
The phrase "of students enrolled in a language course" tells you which group you should be focusing on for the denominator. Which column of the table is that?
Choose the correct denominator
Add the appropriate numbers in the "Enrolled in a language course" column to find how many students are in that group in total. That total is your denominator.
Find the intersection for the numerator
Within the language group, which cell shows the students who are also enrolled in a science course? That number will be your numerator over the total language students.
Turn the fraction into a percent
Once you have the correct fraction (students in both divided by all language students), convert it to a decimal and then multiply by 100 to express it as a percent. Don’t forget to round to the nearest tenth of a percent.
Desmos Guide
Compute the conditional fraction
In a Desmos expression line, type 2700/(2700+3900) and observe the decimal value that appears; this is the fraction of language students who also take science.
Convert to a percent and round
In another expression line, type 2700/(2700+3900)*100 to see this value as a percent. Use that output, rounded to the nearest tenth, as your final answer.
Step-by-step Explanation
Interpret the question as a conditional percent
The question asks: "what percent of students enrolled in a language course are also enrolled in a science course?"
This is a conditional percent:
- We are looking at only the students enrolled in a language course (this is our group, or denominator).
- Within that group, we want the fraction who are also enrolled in a science course (this is our numerator).
Find the total number of language students (denominator)
Look at the column "Enrolled in a language course."
- Enrolled in a science course and language:
- Not enrolled in a science course but enrolled in language:
So the total number of students who are enrolled in a language course is
This will be the denominator of our fraction.
Find how many are in both language and science (numerator)
From the table, the number of students enrolled in both a language course and a science course is the top-left cell: .
So the fraction of language students who also take science is
We can simplify this fraction by dividing numerator and denominator by :
In the next step, we will turn this fraction into a percent.
Convert the fraction to a percent and round
Convert to a decimal by dividing by (you can use a calculator):
- (a little more than ).
Now change the decimal to a percent by multiplying by :
So, to the nearest tenth of a percent, of students enrolled in a language course are also enrolled in a science course. This corresponds to answer choice C) 40.9%.