Let's break down the data from the Venn diagram and use it to answer the questions.

Total number of dancers:

72 students.

Venn Diagram Data Breakdown:

• Smooth only: 10 dancers
• Rhythm only: 4 dancers
• Standard only: 14 dancers
• Smooth and Rhythm (but not Standard): 6 dancers
• Smooth and Standard (but not Rhythm): 12 dancers
• Rhythm and Standard (but not Smooth): 7 dancers
• All three categories (Smooth, Rhythm, and Standard): 8 dancers

Probability Calculations:

Let's go question by question:

a) The dancer has competed in the Smooth category:

The total number of dancers in the Smooth category includes:

• Smooth only: 10
• Smooth and Rhythm: 6
• Smooth and Standard: 12
• All three categories: 8

So, the total number of dancers in the Smooth category = $10 + 6 + 12 + 8 = 36$.

The probability of selecting a dancer who competed in Smooth is: $P(\text{Smooth}) = \frac{36}{72} = 0.5000$

b) The dancer has competed in the Rhythm and Standard categories:

The dancers who have competed in both Rhythm and Standard are:

• Rhythm and Standard only: 7
• All three categories: 8

Total number of dancers in both Rhythm and Standard = $7 + 8 = 15$.

The probability of selecting a dancer who competed in both Rhythm and Standard is: $P(\text{Rhythm and Standard}) = \frac{15}{72} \approx 0.2083$

c) The dancer has competed in the Smooth or Standard categories:

The total number of dancers who competed in Smooth or Standard includes:

• Smooth only: 10
• Standard only: 14
• Smooth and Rhythm: 6
• Smooth and Standard: 12
• Rhythm and Standard: 7
• All three categories: 8

Total number of dancers in Smooth or Standard = $10 + 14 + 6 + 12 + 7 + 8 = 57$.

The probability of selecting a dancer who competed in Smooth or Standard is: $P(\text{Smooth or Standard}) = \frac{57}{72} \approx 0.7917$

d) The dancer has competed in the Standard category, given that they have competed in the Rhythm category:

First, we need to know how many dancers have competed in Rhythm. The total number of Rhythm dancers includes:

• Rhythm only: 4
• Smooth and Rhythm: 6
• Rhythm and Standard: 7
• All three categories: 8

Total number of Rhythm dancers = $4 + 6 + 7 + 8 = 25$.

Now, how many of these dancers also competed in the Standard category? Those who competed in both Rhythm and Standard are:

• Rhythm and Standard: 7
• All three categories: 8

Total number of Rhythm dancers who also competed in Standard = $7 + 8 = 15$.

The conditional probability is: $P(\text{Standard} \mid \text{Rhythm}) = \frac{15}{25} = 0.6000$

Would you like any further details or have any questions on these calculations?

5 Follow-up Questions:

1. What is the probability that a randomly selected dancer has competed in all three categories?
2. What is the probability that a dancer has competed in exactly one category?
3. What is the probability that a dancer has competed in at least two categories?
4. How many dancers have competed in only Smooth or Rhythm?
5. What is the conditional probability that a dancer has competed in Smooth, given that they have competed in Standard?

Tip:

To calculate "or" probabilities involving categories, always account for overlaps to avoid double-counting those who belong to multiple categories.