SOLUTION: At a liberal arts college in the Midwest, 39% of freshman are enrolled in both a math course and an English course, 11% are enrolled in a math course, but not an English course, an

Question 1201039: At a liberal arts college in the Midwest, 39% of freshman are enrolled in both a math course and an English course, 11% are enrolled in a math course, but not an English course, and 63% are enrolled in an English course. What is the probability that a freshman student selected at random is taking an English course but not a mathematics course? What is the probability a student is not enrolled in either course?
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

This is one way to have a starting Venn Diagram template.

M = set of freshmen taking math (some also taking English)
E = set of freshmen taking English (some also taking math)
region a = freshmen taking math only
region b = freshmen taking both math and English
region c = freshmen taking English only
region d = freshmen taking neither course

Let's say there are 1000 freshmen as a hypothetical situation.

Given facts

Number	Statement
1	39% of freshmen are enrolled in both a math course and an English course
2	11% are enrolled in a math course, but not an English course
3	63% are enrolled in an English course

Fact 1 means there are 0.39*1000 = 390 freshmen enrolled in both math and English.
This value will replace "b" in the Venn Diagram above.

Fact 2 means there are 0.11*1000 = 110 freshmen enrolled in math, but not in English.
This value will replace "a" in the Venn Diagram above.

Fact 3 means there are 0.63*1000 = 630 freshmen enrolled in English (some of these students may also be in math).
Subtract off the amount of freshmen taking both to get 630-390 = 240.
This is the number of freshmen taking English only.
This value replaces the "c" in the Venn Diagram above.

This is what you should have so far.

The a,b,c was replaced with 110, 390, and 240 in that exact order.
The amount of freshmen so far is 110+390+240 = 740

Since there are 1000 freshmen total, we know that 1000-740 = 260 freshmen aren't taking math and aren't English either.
This value will replace "d" in the diagram.

Here's the final Venn Diagram

As a check:
a+b+c+d = 110+390+240+260 = 1000

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Now to answer the questions.

Problem 1) What is the probability that a freshman student selected at random is taking an English course but not a mathematics course?

Refer to the Venn Diagram above.
We see there are 240 freshmen taking English but not math.
This is out of 1000 freshmen total.
240/1000 = 0.24

Answer: 0.24

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Problem 2) What is the probability a student is not enrolled in either course?

Refer to the Venn Diagram above.
There are 260 freshmen that aren't taking either course mentioned.
This is out of 1000 freshmen total.

260/1000 = 0.26

Answer: 0.24

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