Question

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 passed the test given that they completed the homework?

	passed the test	failed the test
did not complete the homework	2	6

Explanation:

Step1: Recall Conditional Probability Formula

The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). In this case, event \( A \) is "passed the test" and event \( B \) is "completed the homework". So we need to find the number of students who passed the test and completed the homework (\( A \cap B \)) and the number of students who completed the homework (\( B \)).

Step2: Find Number of Students in \( A \cap B \) and \( B \)

From the table, the number of students who completed the homework and passed the test (\( A \cap B \)) is 9. The number of students who completed the homework (\( B \)) is the sum of those who completed homework and passed, and those who completed homework and failed: \( 9 + 4 = 13 \).

Step3: Calculate the Probability

Using the conditional probability formula, \( P(\text{passed} | \text{completed homework}) = \frac{\text{Number of students who completed homework and passed}}{\text{Number of students who completed homework}} = \frac{9}{13} \).

Answer:

\(\frac{9}{13}\)