https://file.aiquickdraw.com/mathbot/file/1730256067182ljx37urv.jpg

Let's solve the questions given in the image.

### Given:
- \( P(A) = \frac{1}{4} \)
- \( P(B) = \frac{1}{5} \)
- \( P(A \cap B) = \frac{1}{7} \)

### Questions:

1. **Find \( P(A \cup B) \):**

   The formula for the union of two events \( A \) and \( B \) is:
   \[
   P(A \cup B) = P(A) + P(B) - P(A \cap B)
   \]
   Plugging in the values:
   \[
   P(A \cup B) = \frac{1}{4} + \frac{1}{5} - \frac{1}{7}
   \]
   We need a common denominator to perform the addition and subtraction. The least common multiple of 4, 5, and 7 is 140.

   \[
   P(A \cup B) = \frac{35}{140} + \frac{28}{140} - \frac{20}{140} = \frac{43}{140}
   \]

   So, \( P(A \cup B) = \frac{43}{140} \).

2. **Find \( P(A | B) \):**

   The formula for the conditional probability \( P(A | B) \) is:
   \[
   P(A | B) = \frac{P(A \cap B)}{P(B)}
   \]
   Substituting the values:
   \[
   P(A | B) = \frac{\frac{1}{7}}{\frac{1}{5}} = \frac{1}{7} \times \frac{5}{1} = \frac{5}{7}
   \]

   Therefore, \( P(A | B) = \frac{5}{7} \).

3. **Find \( P(B | A^c) \):**

   Here, \( A^c \) denotes the complement of \( A \). The formula for \( P(B | A^c) \) is:
   \[
   P(B | A^c) = \frac{P(B \cap A^c)}{P(A^c)}
   \]
   We know that:
   \[
   P(A^c) = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4}
   \]

   To find \( P(B \cap A^c) \), we use:
   \[
   P(B) = P(B \cap A) + P(B \cap A^c)
   \]
   Rearranging, we get:
   \[
   P(B \cap A^c) = P(B) - P(A \cap B) = \frac{1}{5} - \frac{1}{7}
   \]

   Finding a common denominator (35):
   \[
   P(B \cap A^c) = \frac{7}{35} - \frac{5}{35} = \frac{2}{35}
   \]

   Thus,
   \[
   P(B | A^c) = \frac{\frac{2}{35}}{\frac{3}{4}} = \frac{2}{35} \times \frac{4}{3} = \frac{8}{105}
   \]

   So, \( P(B | A^c) = \frac{8}{105} \).

### Summary of Answers:

- **a. \( P(A \cup B) = \frac{43}{140} \)**
- **b. \( P(A | B) = \frac{5}{7} \)**
- **c. \( P(B | A^c) = \frac{8}{105} \)**

Would you like more detailed explanations for each step, or do you have any specific questions?

---

### Further Related Questions:

1. How do you calculate \( P(A^c \cup B) \)?
2. What would \( P(B^c | A) \) represent, and how is it calculated?
3. How would the answer change if \( P(A \cap B) \) were different?
4. Can you explain the difference between \( P(A \cap B) \) and \( P(A | B) \)?
5. What is the formula for conditional probability when events are independent?

### Tip:
When solving probability problems, remember to always check if events are independent, as it can simplify calculations.

Diketahui P(A) = 1/4, P(B) = 1/5, P(A ∩ B) = 1/7. Tentukan P(A ∪ B), P(A | B), dan P(B | A^c).

Learn how to solve probability questions involving the union of events, conditional probability, and complements. Given P(A) = 1/4, P(B) = 1/5, and P(A ∩ B) = 1/7, find P(A ∪ B), P(A | B), and P(B | A^c). Suitable for high school students (Grades 10-12).

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