**A survey was conducted at a local ballroom dance studio asking students what dances they competed in at the most recent competition. Below are the results for the dances Bachata,FoxTrot, and Bolero:** **297 just bachata 272 bachata and foxtrot 289 just foxtrot 360 bachata and bolero 312 foxtrot and bolero 330 just bolero 322 for all three and 352 for none of them** **If a student is chosen at random, what is the probability that:** *Round all answers to 3 decimal places as needed.* **a) The student competed in Bachata, FoxTrot, or Bolero?**

**b) The student competed in none of these three dances?**

**c) The student competed in FoxTrot & Bolero, but not Bachata?**

**d) The student competed in Bolero, given they competed in Bachata?**

**e) The student competed in Bachata or Bolero, but not FoxTrot?>**

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.

A survey was conducted at a local ballroom dance studio asking students what dances they competed in at the most recent competition. Below are the results for the dances Bachata, FoxTrot, and Bolero: 297 just bachata 272 bachata and foxtrot 289 just foxtrot 360 bachata and bolero 312 foxtrot and bolero 330 just bolero 322 for all three and 352 for none of them. If a student is chosen at random, what is the probability that: a) The student competed in Bachata, FoxTrot, or Bolero? b) The student competed in none of these three dances? c) The student competed in FoxTrot & Bolero, but not Bachata? d) The student competed in Bolero, given they competed in Bachata? e) The student competed in Bachata or Bolero, but not FoxTrot?

Solve probability questions based on survey data from a local ballroom dance studio. Find the probabilities of students competing in Bachata, FoxTrot, or Bolero using set theory and conditional probability. Includes detailed step-by-step solutions. Suitable for high school students in Grades 10-12.

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