To find the probability that a student competed in Bachata or Bolero, but not FoxTrot, we'll follow these steps:

Step 1: Identify the regions in the Venn diagram for Bachata or Bolero but not FoxTrot.

• The Bachata set consists of the numbers 297, 360, and 272 (overlap with FoxTrot).
• The Bolero set consists of the numbers 330, 360, and 322 (overlap with FoxTrot).
• We exclude the regions where FoxTrot is involved, meaning we remove all numbers that intersect with FoxTrot (i.e., 272, 322, 312, and 289).

Thus, the relevant regions are:

• Bachata only: 297
• Bolero only: 330
• Bachata & Bolero (but not FoxTrot): 360

Step 2: Total students in Bachata or Bolero, but not FoxTrot.

Add the numbers from Step 1: $297 \, \text{(Bachata only)} + 330 \, \text{(Bolero only)} + 360 \, \text{(Bachata and Bolero only)} = 987$

Step 3: Total number of students.

To calculate the total number of students, sum up all the regions in the Venn diagram: $297 + 360 + 330 + 272 + 322 + 289 + 312 + 352 = 2534$

Step 4: Probability calculation.

The probability that a student competed in Bachata or Bolero, but not FoxTrot is the ratio of the relevant students to the total number of students: $P = \frac{987}{2534} \approx 0.389$

Thus, the probability is approximately 0.389.

Do you have any questions or want more details? Here are some related questions you might find useful:

1. What is the probability that a student competed in all three events (Bachata, Bolero, and FoxTrot)?
2. What is the probability that a student competed in exactly two events?
3. What is the probability that a student competed in only FoxTrot?
4. What is the probability that a student did not compete in Bolero?
5. What is the probability that a student competed in Bachata and Bolero?

Tip: Always ensure that you're counting disjoint regions correctly in a Venn diagram to avoid double-counting overlapping sections.