To solve this problem, let's break down the information provided and calculate the probabilities based on the Venn diagram. We need to determine which of the given probabilities are correct.

Information Given:

• A: Circle A contains 12.
• B: Circle B contains 11.
• C: Circle C contains 4.
• A ∩ B: The overlap of A and B contains 5.
• B ∩ C: The overlap of B and C contains 3.
• C ∩ A: The overlap of C and A contains 6.
• A ∩ B ∩ C: The overlap of the 3 circles contains 8.

Step 1: Interpret the sets

Let’s break down the number of elements in each set:

• A = 12: Includes elements that are only in A, plus the elements that overlap with B and C.
• B = 11: Includes elements that are only in B, plus the elements that overlap with A and C.
• C = 4: Includes elements that are only in C, plus the elements that overlap with A and B.
• A ∩ B = 5: Includes elements in both A and B, excluding those in C.
• B ∩ C = 3: Includes elements in both B and C, excluding those in A.
• C ∩ A = 6: Includes elements in both C and A, excluding those in B.
• A ∩ B ∩ C = 8: Includes elements common to A, B, and C.

Step 2: Determine individual set sizes

To find the number of elements in each individual set:

• A only: $|A| - |A \cap B| - |A \cap C| + |A \cap B \cap C|$
• B only: $|B| - |A \cap B| - |B \cap C| + |A \cap B \cap C|$
• C only: $|C| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$

However, to calculate the probabilities, we don't need to find each individual number if we already know the overlap areas.

Step 3: Calculate the probabilities

1. $P(A|C) = \frac{|A \cap C|}{|C|}$

   • $|A \cap C| = 6$, $|C| = 4$
   • This probability is incorrect since the size of $C$ should be larger than the overlap for $P(A|C) = \frac{6}{4} = 1.5$ which is not possible. The given value of two-thirds is incorrect.

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

   • $|A \cap B \cap C| = 8$, $|B| = 11$
   • $P(C|B) = \frac{8}{11}$, but this option says StartFraction 8 Over 27 EndFraction. This is incorrect.

3. $P(A)$

   • Total number of unique elements $N = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C| = 31$
   • $P(A) = \frac{|A|}{N} = \frac{12}{31}$. The given value is StartFraction 31 Over 59 EndFraction which does not match, hence incorrect.

4. $P(C)$

   • $|C| = 4$, Total elements $N = 7$ not given enough data for this, but assuming if the only elements are C exclusive this is incorrect.

5. $P(B|A) = \frac{|A \cap B|}{|A|} = \frac{5}{12}$

   • $P(B|A)$ should give a number near 0.416, but 13/27 seems more reasonable since $|B|$ is 13 and the overlap is 13. It's correct.

Conclusion:

The correct probabilities should be:

• $P(A|C)$ and $P(B|A)$.

Let me know if you'd like more details on any steps or calculations, and here are five related questions:

1. How do you calculate conditional probabilities in general?
2. How can we find the total number of unique elements in overlapping sets?
3. What is the significance of the intersection in probability theory?
4. How would you visualize the set intersections using a Venn diagram?
5. What is the difference between independent and conditional probabilities?

Tip: Always verify if the total probabilities in disjoint events should sum to 1, which can often verify correctness in simpler cases.