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To solve the problem, we need to find the conditional probability that a student competed in Jive, given that they competed in Viennese Waltz.

### Conditional Probability Formula:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
Where:
- \( A \) is the event that a student competed in Jive.
- \( B \) is the event that a student competed in Viennese Waltz.

From the Venn diagram:
- The number of students who competed in both Jive and Viennese Waltz (\( A \cap B \)) is the sum of the students in the overlap of both Jive and Viennese Waltz. This includes:
  - 210 students who competed in all three dances.
  - 373 students who competed in only Viennese Waltz and Jive.
  
  So, \( P(A \cap B) = 210 + 373 = 583 \).

- The total number of students who competed in Viennese Waltz (\( B \)) is the sum of all students in the Viennese Waltz circle:
  - 223 (only Viennese Waltz),
  - 390 (Viennese Waltz and Bolero),
  - 373 (Viennese Waltz and Jive),
  - 210 (all three dances).

  Thus, \( P(B) = 223 + 390 + 373 + 210 = 1196 \).

### Now, calculate the conditional probability:
\[
P(\text{Jive} | \text{Viennese Waltz}) = \frac{583}{1196}
\]

Let's compute this value.The probability that a student competed in Jive, given that they competed in Viennese Waltz, is approximately **0.487** (rounded to three decimal places).

Would you like further details or have any questions?

Here are some related questions to expand on this topic:
1. What is the probability that a student competed in all three dances?
2. What is the probability that a student competed in Bolero, given they competed in Viennese Waltz?
3. How many students competed in exactly two dances?
4. What is the probability that a randomly chosen student competed in only Jive?
5. How would the probabilities change if an additional category was added for a fourth dance?

**Tip:** Conditional probabilities are useful for refining predictions based on known information!

The student competed in Jive, given they competed in Viennese Waltz?

Learn how to calculate the conditional probability that a student competed in Jive, given that they participated in Viennese Waltz. Using the Venn diagram data from a ballroom dance competition, we explore key probability concepts and the law of conditional probability. Suitable for high school students.

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