Question
Five years after 650 high school seniors graduate, 400 had a college degree and 310 were married. Half of the students with a college degree were married. What is the probability that a student does not have a college degree or is married?
Show all work and round all decimals to 3 places or percents to the nearest tenth, if needed. (Hint: Try making a two-way table or Venn diagram to organize your information.)
Show all work and round all decimals to 3 places or percents to the nearest tenth, if needed. (Hint: Try making a two-way table or Venn diagram to organize your information.)
Answers
GPT-4o mini
To find the probability that a student does not have a college degree or is married, we can use the principle of complementary events and organize the information in a way that allows us to apply the formulas easily. Let's first summarize the information given:
- Total students = 650
- Students with college degrees = 400
- Students who are married = 310
- Students who have a college degree and are married = \( \frac{1}{2} \times 400 = 200 \)
### Step 1: Organize the Information
Let's set:
- \( A \): the event that a student has a college degree.
- \( B \): the event that a student is married.
We have the following data:
- \( |A| = 400 \) (students with a college degree)
- \( |B| = 310 \) (students who are married)
- \( |A \cap B| = 200 \) (students who are both married and have a college degree)
### Step 2: Calculate Students in Each Category
We can calculate the number of students in different categories using the formula for unions of two sets:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
Calculating \( |A \cup B| \):
\[
|A \cup B| = 400 + 310 - 200 = 510
\]
### Step 3: Find \( |A^c \cap B^c| \)
Next, we determine the number of students that do **not** have a college degree (\( A^c \)) or are not married (\( B^c \)).
The total number of students is 650, so the number of students that either have a college degree or are married is \( |A \cup B| = 510 \).
Thus, the number of students that do not have a college degree or are not married is:
\[
|A^c \cap B^c| = 650 - |A \cup B| = 650 - 510 = 140
\]
### Step 4: Calculate the Probability
The probability that a student does not have a college degree or is married is:
\[
P(A^c \cup B^c) = \frac{|A^c \cap B^c|}{\text{Total number of students}} = \frac{140}{650}
\]
Now, we simplify this fraction:
To simplify \( \frac{140}{650} \):
\[
\frac{140 \div 10}{650 \div 10} = \frac{14}{65}
\]
Calculating the decimal:
\[
\frac{14}{65} \approx 0.215384615
\]
Rounding to three decimal places:
\[
P(A^c \cup B^c) \approx 0.215
\]
### Step 5: Presenting the Result
Thus, the probability that a student does not have a college degree or is married is approximately:
\[
\boxed{0.215}
\]
or, as a percentage (rounded to the nearest tenth):
\[
21.5\%
\]
This result indicates that about 21.5% of the students do not have a college degree or are married.
- Total students = 650
- Students with college degrees = 400
- Students who are married = 310
- Students who have a college degree and are married = \( \frac{1}{2} \times 400 = 200 \)
### Step 1: Organize the Information
Let's set:
- \( A \): the event that a student has a college degree.
- \( B \): the event that a student is married.
We have the following data:
- \( |A| = 400 \) (students with a college degree)
- \( |B| = 310 \) (students who are married)
- \( |A \cap B| = 200 \) (students who are both married and have a college degree)
### Step 2: Calculate Students in Each Category
We can calculate the number of students in different categories using the formula for unions of two sets:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
Calculating \( |A \cup B| \):
\[
|A \cup B| = 400 + 310 - 200 = 510
\]
### Step 3: Find \( |A^c \cap B^c| \)
Next, we determine the number of students that do **not** have a college degree (\( A^c \)) or are not married (\( B^c \)).
The total number of students is 650, so the number of students that either have a college degree or are married is \( |A \cup B| = 510 \).
Thus, the number of students that do not have a college degree or are not married is:
\[
|A^c \cap B^c| = 650 - |A \cup B| = 650 - 510 = 140
\]
### Step 4: Calculate the Probability
The probability that a student does not have a college degree or is married is:
\[
P(A^c \cup B^c) = \frac{|A^c \cap B^c|}{\text{Total number of students}} = \frac{140}{650}
\]
Now, we simplify this fraction:
To simplify \( \frac{140}{650} \):
\[
\frac{140 \div 10}{650 \div 10} = \frac{14}{65}
\]
Calculating the decimal:
\[
\frac{14}{65} \approx 0.215384615
\]
Rounding to three decimal places:
\[
P(A^c \cup B^c) \approx 0.215
\]
### Step 5: Presenting the Result
Thus, the probability that a student does not have a college degree or is married is approximately:
\[
\boxed{0.215}
\]
or, as a percentage (rounded to the nearest tenth):
\[
21.5\%
\]
This result indicates that about 21.5% of the students do not have a college degree or are married.