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**If a student is chosen at random, what is the probability that:**
*Round all answers to 3 decimal places as needed.*
**d) The student competed in Bolero, given they competed in Bachata?**

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.

If a student is chosen at random, what is the probability that the student competed in Bolero, given they competed in Bachata?

This problem explores the conditional probability of a student competing in Bolero given they competed in Bachata. It covers key probability concepts and uses a Venn diagram for visualization. The solution applies the conditional probability formula: P(Bolero | Bachata) = P(Bolero ∩ Bachata) / P(Bachata). Suitable for high school students in Grades 10-12.

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