Let's determine the probability that a randomly chosen student from the class either passed the test or completed the homework. We will proceed step-by-step:
1. Identify and sum up the total number of students:
- Students who passed the test and completed the homework: 12
- Students who failed the test and completed the homework: 6
- Students who passed the test and did not complete the homework: 3
- Students who failed the test and did not complete the homework: 4
Total number of students = [tex]\(12 + 6 + 3 + 4 = 25\)[/tex]
2. Identify the number of students who passed the test:
- Students who passed the test and completed the homework: 12
- Students who passed the test and did not complete the homework: 3
Number of students who passed the test = [tex]\(12 + 3 = 15\)[/tex]
3. Identify the number of students who completed the homework:
- Students who passed the test and completed the homework: 12
- Students who failed the test and completed the homework: 6
Number of students who completed the homework = [tex]\(12 + 6 = 18\)[/tex]
4. Determine the number of students who either passed the test or completed the homework:
Using the principle of inclusion and exclusion:
[tex]\[
\text{Number of students who either passed or completed} = (\text{Number who passed}) + (\text{Number who completed}) - (\text{Number who did both})
\][/tex]
Number of students who did both (passed the test and completed the homework): 12
[tex]\[
\text{Number who either passed or completed} = 15 + 18 - 12 = 21
\][/tex]
5. Calculate the probability:
Probability of choosing a student who either passed the test or completed the homework:
[tex]\[
\text{Probability} = \frac{\text{Number who either passed or completed}}{\text{Total number of students}} = \frac{21}{25}
\][/tex]
Simplifying the fraction if necessary:
[tex]\[
\frac{21}{25} = 0.84
\][/tex]
So, the probability that a randomly chosen student from the class either passed the test or completed the homework is [tex]\( \frac{21}{25} \)[/tex] or 0.84.
