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Let's break down the contingency table and solve the given problems:

### Contingency Table:

|                                | Completed Homework | Did Not Complete Homework | Total |
|---------------------------------|--------------------|---------------------------|-------|
| **Passed the Course**           | 25                 | 14                        | 39    |
| **Did Not Pass the Course**     | 3                  | 7                         | 10    |
| **Total**                       | 28                 | 21                        | 49    |

### Questions:

1. **What is the probability a student passed the course given that they completed their homework?**

   Formula:  
   \[
   P(\text{Passed} \mid \text{Completed Homework}) = \frac{P(\text{Passed and Completed Homework})}{P(\text{Completed Homework})}
   \]

   - Number of students who passed and completed homework = 25.
   - Total number of students who completed homework = 28.

   So,  
   \[
   P(\text{Passed} \mid \text{Completed Homework}) = \frac{25}{28} \approx 0.89286 \quad \text{or} \quad 89.29\%.
   \]

2. **What is the probability a student passed the course given that they didn't complete their homework?**

   Formula:  
   \[
   P(\text{Passed} \mid \text{Did Not Complete Homework}) = \frac{P(\text{Passed and Did Not Complete Homework})}{P(\text{Did Not Complete Homework})}
   \]

   - Number of students who passed and did not complete homework = 14.
   - Total number of students who did not complete homework = 21.

   So,  
   \[
   P(\text{Passed} \mid \text{Did Not Complete Homework}) = \frac{14}{21} = 0.66667 \quad \text{or} \quad 66.67\%.
   \]

3. **What is the probability a student completed homework given that they passed the course?**

   Formula:  
   \[
   P(\text{Completed Homework} \mid \text{Passed}) = \frac{P(\text{Passed and Completed Homework})}{P(\text{Passed})}
   \]

   - Number of students who passed and completed homework = 25.
   - Total number of students who passed = 39.

   So,  
   \[
   P(\text{Completed Homework} \mid \text{Passed}) = \frac{25}{39} \approx 0.64103 \quad \text{or} \quad 64.10\%.
   \]

4. **What is the probability a student didn’t complete homework given that they didn’t pass the course?**

   Formula:  
   \[
   P(\text{Did Not Complete Homework} \mid \text{Did Not Pass}) = \frac{P(\text{Did Not Pass and Did Not Complete Homework})}{P(\text{Did Not Pass})}
   \]

   - Number of students who didn’t pass and didn’t complete homework = 7.
   - Total number of students who didn’t pass = 10.

   So,  
   \[
   P(\text{Did Not Complete Homework} \mid \text{Did Not Pass}) = \frac{7}{10} = 0.7 \quad \text{or} \quad 70\%.
   \]

---

### Summary of Results:

1. \( P(\text{Passed} \mid \text{Completed Homework}) = 89.29\% \)
2. \( P(\text{Passed} \mid \text{Did Not Complete Homework}) = 66.67\% \)
3. \( P(\text{Completed Homework} \mid \text{Passed}) = 64.10\% \)
4. \( P(\text{Did Not Complete Homework} \mid \text{Did Not Pass}) = 70.00\% \)

---

Would you like further details on any step? Here are some related questions you might want to explore:

1. What is the probability a student did not pass the course given that they completed homework?
2. What is the probability a student passed the course regardless of homework completion?
3. How would you calculate the probability of not completing homework based on total students?
4. How can these probabilities be visualized in a Venn diagram or tree diagram?
5. How does conditional probability differ from joint probability?

**Tip**: Conditional probability involves narrowing down your sample space to a subset of interest—it's always important to identify which event is the condition and which is the outcome.

Contingency table and conditional probability problems involving students passing or failing a course based on homework completion.

Explore conditional probability problems using contingency tables. This example involves the likelihood of students passing or failing a course based on whether they completed their homework. Detailed solutions provided for questions like 'What is the probability a student passed given that they completed homework?'

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