To solve the problem using a Venn diagram approach, we will first organize the given data and calculate the total number of students. Then we can calculate the requested probabilities step by step.

Given Data:

• $|B \cap F' \cap B'| = 297$ (Only Bachata)
• $|B \cap F| = 272$ (Bachata and Foxtrot)
• $|F \cap B' \cap B'| = 289$ (Only Foxtrot)
• $|B \cap B| = 360$ (Bachata and Bolero)
• $|F \cap B| = 312$ (Foxtrot and Bolero)
• $|B' \cap F' \cap B| = 330$ (Only Bolero)
• $|B \cap F \cap B| = 322$ (All three dances)
• $|B' \cap F' \cap B'| = 352$ (None of the dances)

Let:

• $B$ represent the set of students who competed in Bachata,
• $F$ represent the set of students who competed in Foxtrot,
• $Bo$ represent the set of students who competed in Bolero.

First, calculate the total number of students.

1. Total number of students:

The total number of students is the sum of all mutually exclusive groups: $\text{Total} = 297 + 272 + 289 + 360 + 312 + 330 + 322 + 352$ Let’s calculate this total.

The total number of students is $2534$.

Now, let's calculate each of the probabilities as requested:

a) Probability that a student competed in Bachata, FoxTrot, or Bolero

This is the probability that a student competed in at least one of the three dances, which can be calculated as: $P(\text{Bachata, Foxtrot, or Bolero}) = 1 - P(\text{None of them})$ Thus, we will subtract the students who did not compete in any of the dances from the total.

$P(\text{Bachata, Foxtrot, or Bolero}) = \frac{\text{Total} - 352}{\text{Total}}$

b) Probability that a student competed in none of these three dances

This is simply: $P(\text{None}) = \frac{352}{2534}$

c) Probability that a student competed in FoxTrot & Bolero, but not Bachata

We want the students who competed in both FoxTrot and Bolero but not Bachata, which is given by: $P(\text{FoxTrot} \cap \text{Bolero} \cap \text{not Bachata}) = \frac{\text{Foxtrot and Bolero} - \text{All three}}{\text{Total}}$ This is: $P(\text{FoxTrot} \cap \text{Bolero} \cap \text{not Bachata}) = \frac{312 - 322}{2534}$

d) Probability that a student competed in Bolero, given they competed in Bachata

This is the conditional probability: $P(\text{Bolero} \mid \text{Bachata}) = \frac{P(\text{Bachata and Bolero})}{P(\text{Bachata})}$ Where $P(\text{Bachata})$ includes all students who participated in Bachata.

e) Probability that a student competed in Bachata or Bolero, but not FoxTrot

We want the students who competed in Bachata or Bolero, excluding FoxTrot: $P((\text{Bachata} \cup \text{Bolero}) \cap \text{not Foxtrot}) = \frac{\text{Only Bachata} + \text{Only Bolero} + \text{Bachata and Bolero (not Foxtrot)}}{\text{Total}}$

Now, let's calculate these probabilities.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through the steps to calculate the answers manually.

Would you like further assistance or details on how to calculate any of the parts?

Here are some related questions you might find helpful:

1. How do we calculate the total probability for three events using a Venn diagram?
2. How is conditional probability used in real-world scenarios?
3. What are the relationships between intersections and unions in set theory?
4. How does the inclusion-exclusion principle apply to this problem?
5. What are the steps to calculate probabilities from survey data?

Tip: Always start with visualizing the problem using a Venn diagram to simplify the overlapping sets.