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 is not married or does not have a college degree?
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 is not married or does not have a college degree, we can use a Venn diagram or a two-way table to organize our information. Let's denote:
- \( A \): students with a college degree
- \( B \): students who are married
From the problem, we know:
- Total students = 650
- Students with a college degree = 400
- Students who are married = 310
- Students with both a college degree and married = half of those with a college degree = \( 0.5 \times 400 = 200 \)
### Step 1: Organize Information
- Let \( |A| = 400 \) (students with a college degree)
- Let \( |B| = 310 \) (students who are married)
- Let \( |A \cap B| = 200 \) (students with both a college degree and married)
### Step 2: Use the Inclusion-Exclusion Principle
The number of students who have either a college degree or are married (or both) can be calculated using the following formula:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
Substituting the known values:
\[
|A \cup B| = 400 + 310 - 200 = 510
\]
So, 510 students have either a college degree, are married, or both.
### Step 3: Determine Those Who Are Not Married or Do Not Have a Degree
We need to find those who are not married or do not have a college degree. First, we can find those who are neither married nor have a college degree by subtracting this from the total number of students:
\[
\text{Students not married and not having a degree} = \text{Total students} - |A \cup B|
\]
Calculating:
\[
\text{Students not married and not having a degree} = 650 - 510 = 140
\]
### Step 4: Calculate the Probability
The probability that a student is not married or does not have a college degree can be calculated by dividing the number of students not married or without a degree by the total number of students:
\[
P(\text{not married or not a degree}) = \frac{\text{Students not married and not having a degree}}{\text{Total students}}
\]
\[
P(\text{not married or not a degree}) = \frac{140}{650}
\]
### Step 5: Simplifying the Probability
Simplifying this fraction:
\[
P(\text{not married or not a degree}) = \frac{140 \div 10}{650 \div 10} = \frac{14}{65}
\]
Calculating this as a decimal:
\[
\frac{14}{65} \approx 0.21538461538 \approx 0.215
\]
### Final Result
Rounding to three decimal places:
\[
\text{The probability that a student is not married or does not have a college degree is } \boxed{0.215}
\]
or approximately 21.5%.
- \( A \): students with a college degree
- \( B \): students who are married
From the problem, we know:
- Total students = 650
- Students with a college degree = 400
- Students who are married = 310
- Students with both a college degree and married = half of those with a college degree = \( 0.5 \times 400 = 200 \)
### Step 1: Organize Information
- Let \( |A| = 400 \) (students with a college degree)
- Let \( |B| = 310 \) (students who are married)
- Let \( |A \cap B| = 200 \) (students with both a college degree and married)
### Step 2: Use the Inclusion-Exclusion Principle
The number of students who have either a college degree or are married (or both) can be calculated using the following formula:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
Substituting the known values:
\[
|A \cup B| = 400 + 310 - 200 = 510
\]
So, 510 students have either a college degree, are married, or both.
### Step 3: Determine Those Who Are Not Married or Do Not Have a Degree
We need to find those who are not married or do not have a college degree. First, we can find those who are neither married nor have a college degree by subtracting this from the total number of students:
\[
\text{Students not married and not having a degree} = \text{Total students} - |A \cup B|
\]
Calculating:
\[
\text{Students not married and not having a degree} = 650 - 510 = 140
\]
### Step 4: Calculate the Probability
The probability that a student is not married or does not have a college degree can be calculated by dividing the number of students not married or without a degree by the total number of students:
\[
P(\text{not married or not a degree}) = \frac{\text{Students not married and not having a degree}}{\text{Total students}}
\]
\[
P(\text{not married or not a degree}) = \frac{140}{650}
\]
### Step 5: Simplifying the Probability
Simplifying this fraction:
\[
P(\text{not married or not a degree}) = \frac{140 \div 10}{650 \div 10} = \frac{14}{65}
\]
Calculating this as a decimal:
\[
\frac{14}{65} \approx 0.21538461538 \approx 0.215
\]
### Final Result
Rounding to three decimal places:
\[
\text{The probability that a student is not married or does not have a college degree is } \boxed{0.215}
\]
or approximately 21.5%.