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In a class of students, the following data table summarizes how many students passed a test and completed the homework due the day of the test. What is the probability that a student chosen randomly from the class passed the test or completed the homework?

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
& \begin{tabular}{c}
Passed \\
the test
\end{tabular}
& \begin{tabular}{c}
Failed \\
the \\
test
\end{tabular} \\
\hline
\begin{tabular}{c}
Completed the \\
homework
\end{tabular}
& 12
& 6 \\
\hline
\begin{tabular}{c}
Did not complete \\
the homework
\end{tabular}
& 3
& 4 \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnswer
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%).
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