We are asked to find the conditional probability that a randomly selected student competed in Bolero, given that they competed in Bachata. This is a conditional probability problem and can be represented as:

$P(\text{Bolero} | \text{Bachata}) = \frac{P(\text{Bolero} \cap \text{Bachata})}{P(\text{Bachata})}$

Step 1: Calculate $P(\text{Bachata})$

This is the total number of students who competed in Bachata. To find this, we sum all the values in the Bachata circle:

• Students who only competed in Bachata: 297
• Students who competed in Bachata and FoxTrot: 272
• Students who competed in Bachata and Bolero: 360
• Students who competed in all three (Bachata, Bolero, and FoxTrot): 322

So, the total number of students who competed in Bachata is:

$P(\text{Bachata}) = 297 + 272 + 360 + 322$

Step 2: Calculate $P(\text{Bolero} \cap \text{Bachata})$

This is the number of students who competed in both Bolero and Bachata. To find this, we sum the following regions in the diagram:

• Students who competed in both Bachata and Bolero: 360
• Students who competed in all three (Bachata, Bolero, and FoxTrot): 322

Thus, $P(\text{Bolero} \cap \text{Bachata}) = 360 + 322$.

Step 3: Apply the formula

Now we can substitute these values into the conditional probability formula:

$P(\text{Bolero} | \text{Bachata}) = \frac{P(\text{Bolero} \cap \text{Bachata})}{P(\text{Bachata})}$

Let's compute this.The probability that a student competed in Bolero, given that they competed in Bachata, is approximately $0.545$ or 54.5%.

Would you like more details on any part of this solution? Here are five related questions to expand your understanding:

1. What is the probability that a student competed in FoxTrot, given that they competed in Bolero?
2. How many students competed in only one dance style?
3. What is the probability that a randomly selected student competed in all three styles?
4. What is the probability that a student competed in at least two dance styles?
5. How many students competed in exactly two dance styles?

Tip: For conditional probability, always focus on the "given" event first, narrowing down the sample space accordingly.