Answer :
To find the probability that a randomly chosen student from the class either passed the test or completed the homework, let's break down the solution step-by-step.
### Step 1: Determine the Total Number of Students in the Class
We have the following data:
- 12 students passed the test and completed the homework.
- 6 students failed the test but completed the homework.
- 3 students passed the test but did not complete the homework.
- 4 students failed the test and did not complete the homework.
To find the total number of students, we sum all the given values:
[tex]\[ \text{Total students} = 12 + 6 + 3 + 4 = 25 \][/tex]
### Step 2: Determine the Number of Students Who Passed the Test
Students who passed the test include:
- Those who passed and completed the homework (12 students).
- Those who passed but did not complete the homework (3 students).
Thus, the number of students who passed the test is:
[tex]\[ \text{Passed the test} = 12 + 3 = 15 \][/tex]
### Step 3: Determine the Number of Students Who Completed the Homework
Students who completed the homework include:
- Those who passed the test and completed the homework (12 students).
- Those who failed the test but completed the homework (6 students).
Thus, the number of students who completed the homework is:
[tex]\[ \text{Completed homework} = 12 + 6 = 18 \][/tex]
### Step 4: Calculate the Probability of a Student Passing the Test or Completing the Homework
We use the formula for the probability of the union of two events, [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Where:
- [tex]\( P(A) \)[/tex] is the probability that a student passed the test.
- [tex]\( P(B) \)[/tex] is the probability that a student completed the homework.
- [tex]\( P(A \cap B) \)[/tex] is the probability that a student both passed the test and completed the homework.
From the data:
- [tex]\( P(A) = \frac{\text{Passed the test}}{\text{Total students}} = \frac{15}{25} \)[/tex]
- [tex]\( P(B) = \frac{\text{Completed homework}}{\text{Total students}} = \frac{18}{25} \)[/tex]
- [tex]\( P(A \cap B) = \frac{\text{Passed and completed the homework}}{\text{Total students}} = \frac{12}{25} \)[/tex]
So, the probability is:
[tex]\[ P(\text{Passed test or Completed homework}) = \frac{15}{25} + \frac{18}{25} - \frac{12}{25} = \frac{15 + 18 - 12}{25} = \frac{21}{25} = 0.84 \][/tex]
### Final Answer:
The probability that a student chosen randomly from the class passed the test or completed the homework is [tex]\( 0.84 \)[/tex] (or 84%).
### Step 1: Determine the Total Number of Students in the Class
We have the following data:
- 12 students passed the test and completed the homework.
- 6 students failed the test but completed the homework.
- 3 students passed the test but did not complete the homework.
- 4 students failed the test and did not complete the homework.
To find the total number of students, we sum all the given values:
[tex]\[ \text{Total students} = 12 + 6 + 3 + 4 = 25 \][/tex]
### Step 2: Determine the Number of Students Who Passed the Test
Students who passed the test include:
- Those who passed and completed the homework (12 students).
- Those who passed but did not complete the homework (3 students).
Thus, the number of students who passed the test is:
[tex]\[ \text{Passed the test} = 12 + 3 = 15 \][/tex]
### Step 3: Determine the Number of Students Who Completed the Homework
Students who completed the homework include:
- Those who passed the test and completed the homework (12 students).
- Those who failed the test but completed the homework (6 students).
Thus, the number of students who completed the homework is:
[tex]\[ \text{Completed homework} = 12 + 6 = 18 \][/tex]
### Step 4: Calculate the Probability of a Student Passing the Test or Completing the Homework
We use the formula for the probability of the union of two events, [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Where:
- [tex]\( P(A) \)[/tex] is the probability that a student passed the test.
- [tex]\( P(B) \)[/tex] is the probability that a student completed the homework.
- [tex]\( P(A \cap B) \)[/tex] is the probability that a student both passed the test and completed the homework.
From the data:
- [tex]\( P(A) = \frac{\text{Passed the test}}{\text{Total students}} = \frac{15}{25} \)[/tex]
- [tex]\( P(B) = \frac{\text{Completed homework}}{\text{Total students}} = \frac{18}{25} \)[/tex]
- [tex]\( P(A \cap B) = \frac{\text{Passed and completed the homework}}{\text{Total students}} = \frac{12}{25} \)[/tex]
So, the probability is:
[tex]\[ P(\text{Passed test or Completed homework}) = \frac{15}{25} + \frac{18}{25} - \frac{12}{25} = \frac{15 + 18 - 12}{25} = \frac{21}{25} = 0.84 \][/tex]
### Final Answer:
The probability that a student chosen randomly from the class passed the test or completed the homework is [tex]\( 0.84 \)[/tex] (or 84%).
