To solve the probability that a randomly chosen student competed in Bachata or Bolero, but not FoxTrot, we will analyze the Venn diagram carefully.

Steps:

1. Bachata or Bolero, but not FoxTrot means we are looking at the regions that belong to the Bachata and Bolero circles, but exclude the FoxTrot region. From the Venn diagram, these areas are:

   • The Bachata-only section: 297
   • The Bolero-only section: 330
   • The Bachata and Bolero overlap, but excluding FoxTrot: 360

   These numbers are in the regions that belong only to Bachata or Bolero, or their overlap without FoxTrot.

2. Total students in Bachata or Bolero, but not FoxTrot: $297 + 330 + 360 = 987$

3. Total students (from the diagram):

   • Add up all the numbers from the Venn diagram: $297 + 289 + 352 + 360 + 330 + 312 + 272 + 322 = 2534$

4. Probability: $P(\text{Bachata or Bolero but not FoxTrot}) = \frac{987}{2534} \approx 0.389$

Final answer:

The probability that a randomly chosen student competed in Bachata or Bolero, but not FoxTrot is approximately 0.389.

Do you need more details or have any questions?

Here are 5 related questions you can explore:

1. What is the probability a student competed in all three dance types?
2. What is the probability a student competed in only FoxTrot?
3. What is the probability a student competed in both Bachata and FoxTrot?
4. What is the probability a student competed in Bolero or FoxTrot, but not Bachata?
5. What is the probability a student competed in exactly two dances?

Tip: Understanding the structure of Venn diagrams helps break down complex probability problems into simpler regions.