To find the probability that a student failed the test given that they completed the homework, we need to use conditional probability.

The formula for conditional probability is:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Where:

• $A$ is the event that the student failed the test.
• $B$ is the event that the student completed the homework.

From the table:

• The number of students who completed the homework and failed the test is 5.
• The total number of students who completed the homework is $14 + 5 = 19$.

Thus, the conditional probability is:

$P(\text{Failed the test} | \text{Completed the homework}) = \frac{5}{19}$

So, the probability is:

$P = \frac{5}{19} \approx 0.263$

Would you like more details, or do you have any other questions?

Here are some related questions for further exploration:

1. What is the probability that a student passed the test given that they completed the homework?
2. What is the probability that a student failed the test given they did not complete the homework?
3. What is the overall probability that a student failed the test?
4. What is the overall probability that a student completed the homework?
5. What is the total number of students in the class?

Tip: Conditional probability helps in determining the likelihood of an event under the condition that another event has occurred, narrowing down the possible outcomes.