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From the image you provided, I can see the distribution of favorite sports by city in the table and the associated probabilities you're working on. It seems like you're trying to solve a set of probability problems related to the favorite sports data.

Let's focus on part **(c)**, which is marked incorrect:

**P((C ∪ B) ∩ (A ∪ B))**

This can be translated into words as: the probability of people whose favorite sport is either **College Station (C)** or **Baseball (B)**, and also people whose favorite sport is either **San Antonio (A)** or **Baseball (B)**.

### Step-by-Step Solution:

1. **Union of College Station and Baseball (C ∪ B)**: 
   This means we are interested in people who like **either College Station or Baseball** (or both). From the table, we collect all relevant values:
   
   - College Station: All values from the row for College Station: \(7 + 9 + 13 + 10 = 39\).
   - Baseball: All values from the column for Baseball: \(11 + 10 + 9 + 12 + 11 = 53\).

   However, we must **exclude the overlap** between College Station and Baseball (which is already counted twice): the number of people from **College Station whose favorite sport is Baseball**, which is \(9\).

   Thus, \( C ∪ B = 39 + 53 - 9 = 83\).

2. **Union of San Antonio and Baseball (A ∪ B)**:
   This refers to the people who like **either San Antonio or Baseball** (or both):
   
   - San Antonio: All values from the row for San Antonio: \(18 + 12 + 15 + 8 = 53\).
   - Baseball: The total for Baseball (calculated earlier) is \(53\).

   Again, we subtract the overlap between San Antonio and Baseball, which is \(12\).

   So, \( A ∪ B = 53 + 53 - 12 = 94\).

3. **Intersection of (C ∪ B) and (A ∪ B) ((C ∪ B) ∩ (A ∪ B))**:
   Now, we need to find the overlap between these two sets. The only people who are counted in both are those who like **Baseball** because that’s the only common element between the two groups.

   Therefore, the intersection will be the total number of people whose favorite sport is Baseball, which is \(53\).

4. **Total Sample Size**:
   The total number of people surveyed is the sum of all the values in the table. Adding the totals for each city:

   \( (15 + 11 + 13 + 8) + (18 + 10 + 8 + 7) + (7 + 9 + 13 + 10) + (18 + 12 + 15 + 8) + (22 + 11 + 15 + 12) = 242\).

5. **Final Probability**:
   The probability is the number of people who like Baseball (the intersection) divided by the total number of people:

   \[
   P((C ∪ B) ∩ (A ∪ B)) = \frac{53}{242}
   \]

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Let me know if you'd like further clarifications or details on any of the steps!

Here are some related questions you might find helpful:

1. How do you compute the union of two sets in probability?
2. What is the difference between union and intersection in probability?
3. How can overlapping values affect the calculation of union and intersection?
4. What strategies can help avoid double-counting when dealing with sets?
5. How does the total sample size influence probability calculations?

**Tip**: Always check for overlaps between sets when calculating unions to avoid over-counting individuals.

If a surveyed person is selected at random, compute each of the following probabilities: (a) P(D ∩ S^C), (b) P(X ∪ M), (c) P((C ∪ B) ∩ (A ∪ B)), (d) P((D ∩ F) ∪ (M ∩ B)).

Learn how to compute probabilities involving the union, intersection, and complement of sets with an example problem. This problem involves calculating the probability of events across different sports and cities using set theory principles. Suitable for college-level introductory statistics or probability courses.

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