Question

in a math class with 23 students, a test was given the same day that an assignment was due. there were 13 students who passed the test and 14 students who completed the assignment. there were 6 students who failed the test and also did not complete the assignment. what is the probability that a student chosen randomly from the class passed the test or completed the assignment?

Explanation:

Step 1: Calculate number of passed students.

Total students = 23, failed = 13. So $n(P)=23 - 13 = 10$.

Step 2: Calculate number of non - completed students.

Completed = 14, so $n(\overline{C})=23 - 14 = 9$.

Step 3: Identify failed and non - completed.

Given $n(\overline{P}\cap\overline{C}) = 6$.

Step 4: Calculate failed or non - completed.

Using $n(\overline{P}\cup\overline{C})=n(\overline{P})+n(\overline{C})-n(\overline{P}\cap\overline{C})$, we get $13 + 9 - 6 = 16$.

Step 5: Calculate passed and completed.

$n(P\cap C)=23 - 16 = 7$.

Step 6: Calculate passed or completed.

Using $n(P\cup C)=n(P)+n(C)-n(P\cap C)$, we get $10 + 14 - 7 = 17$.

Step 7: Calculate probability.

Probability = $\frac{17}{23}$.

Answer:

$\frac{25}{23}$ (Wait, no, let's correct. First, find the number of students who passed or completed. Let's use the principle of inclusion - exclusion. Let $P$ be the set of students who passed the test, $C$ be the set who completed the assignment. $n(P) = 23 - 13 = 10$? Wait, no, the problem says: total students 23. Students who passed: let's re - read. "a test was given the same day that an assignment was due. There were 13 students who failed the test and 14 students who completed the assignment. There were 6 students who randomly from the class passed the test or completed the assignment?" Wait, no, the question is: "What is the probability that a student chosen randomly from the class passed the test or completed the assignment?"

First, find the number of students who passed the test: total students - failed = $23 - 13 = 10$. Let $P$ be the set of students who passed the test, so $n(P)=10$. Let $C$ be the set of students who completed the assignment, so $n(C) = 14$. The number of students who failed and did not complete: 6. So the number of students who failed or did not complete: 13 (failed) + 6 (did not complete) -? Wait, no, total students = 23. The number of students who passed or completed is total - (failed and did not complete). Failed and did not complete: 6. So $n(P\cup C)=23 - 6 = 17$? Wait, no, let's use inclusion - exclusion. $n(P\cup C)=n(P)+n(C)-n(P\cap C)$. We need to find $n(P\cap C)$. The number of students who failed:13, so passed: $23 - 13 = 10$. The number of students who did not complete: total - completed = $23 - 14 = 9$. But we know that the number of students who failed and did not complete is 6. So by the principle of inclusion - exclusion for the complement: number of students who failed or did not complete = number of failed + number of did not complete - number of failed and did not complete = $13 + 9 - 6 = 16$. Therefore, the number of students who passed and completed (the complement of failed or did not complete) is $23 - 16 = 7$. So $n(P\cap C)=7$. Then $n(P\cup C)=n(P)+n(C)-n(P\cap C)=10 + 14 - 7 = 17$. Therefore, the probability is $\frac{17}{23}$.

Wait, let's do it step by step:

Step 1: Find the number of students who passed the test.

Total students = 23, number of students who failed the test = 13. So the number of students who passed the test, $n(P)=23 - 13 = 10$.

Step 2: Find the number of students who did not complete the assignment.

Number of students who completed the assignment, $n(C)=14$. So the number of students who did not complete the assignment, $n(\overline{C})=23 - 14 = 9$.

Step 3: Find the number of students who failed and did not complete the assignment.

We are given that this number is 6, i.e., $n(\overline{P}\cap\overline{C}) = 6$.

Step 4: Find the number of students who failed or did not complete the assignment (using inclusion - exclusion for the complement sets).

By the formula $n(\overline{P}\cup\overline{C})=n(\overline{P})+n(\overline{C})-n(\overline{P}\cap\overline{C})$, where $n(\overline{P}) = 13$ (failed), $n(\overline{C}) = 9$ (did not complete), and $n(\overline{P}\cap\overline{C}) = 6$ (failed and did not complete). So $n(\overline{P}\cup\overline{C})=13 + 9 - 6 = 16$.

Step 5: Find the number of students who passed and completed the assignment (complement of $\overline{P}\cup\overline{C}$).

Since $n(P\cap C)=23 - n(\overline{P}\cup\overline{C})$, we have $n(P\cap C)=23 - 16 = 7$.

Step 6: Find the number of students who passed or completed the assignment (using inclusion - exclusion for $P$ and $C$).

By the formula $n(P\cup C)=n(P)+n(C)-n(P\cap C)$, substituting $n(P)=10$, $n(C)=14$, and $n(P\cap C)=7$, we get $n(P\cup C)=10 + 14 - 7 = 17$.

Step 7: Calculate the probability.

The probability is $\frac{n(P\cup C)}{23}=\frac{17}{23}$.