https://file.aiquickdraw.com/mathbot/file/1725505505195goylcg1p.jpg

The image contains a Venn diagram showing the overlap between three Halloween activities: *Smooth*, *Rhythm*, and *StandOut*, involving 73 students. Here's a breakdown of the relevant numbers within the Venn diagram for each activity:

- **Smooth:** 12 students
- **Rhythm:** 14 students
- **StandOut:** 10 students
- **Smooth and Rhythm:** 13 students
- **Rhythm and StandOut:** 9 students
- **StandOut and Smooth:** 4 students
- **All three activities:** 7 students

Additionally, the following probabilities are to be calculated:

### Required Calculations:
The task asks for the probability of students participating in various combinations of the activities:
1. **Rhythm only**
2. **Rhythm and StandOut**
3. **Smooth and StandOut**
4. **Smooth or StandOut**

Each of these results should be given as a percentage.

I will now calculate these percentages based on the total number of 73 students.

### 1. **Rhythm only** 
The number of students in *Rhythm only* is the part of the Venn diagram in the Rhythm circle that does not overlap with Smooth or StandOut. This is 14 students.
\[
\text{Probability of Rhythm only} = \frac{14}{73} \times 100
\]

### 2. **Rhythm and StandOut** 
This is the number of students participating in both Rhythm and StandOut, including those in all three categories (since these students are part of both). From the diagram:
- Rhythm and StandOut (9) includes students in all three (7).
So:
\[
\text{Probability of Rhythm and StandOut} = \frac{9}{73} \times 100
\]

### 3. **Smooth and StandOut** 
This is the number of students participating in both Smooth and StandOut, including those in all three categories. From the diagram:
- Smooth and StandOut (4) includes those in all three activities (7).
So:
\[
\text{Probability of Smooth and StandOut} = \frac{4}{73} \times 100
\]

### 4. **Smooth or StandOut** 
This is the total number of students participating in *either* Smooth or StandOut, which includes:
- Smooth only
- StandOut only
- Both Smooth and StandOut
We calculate this as:
\[
\text{Smooth or StandOut} = \frac{\text{Students in Smooth or StandOut}}{73} \times 100
\]

Let me calculate these for you now.# Given values from the Venn diagram and total number of students
total_students = 73

# Number of students in each case
rhythm_only = 14
rhythm_and_standout = 9
smooth_and_standout = 4
smooth_or_standout = (12 + 10 + 4 + 7)  # Includes smooth only, standout only, and both

# Calculate the probabilities
prob_rhythm_only = (rhythm_only / total_students) * 100
prob_rhythm_and_standout = (rhythm_and_standout / total_students) * 100
prob_smooth_and_standout = (smooth_and_standout / total_students) * 100
prob_smooth_or_standout = (smooth_or_standout / total_students) * 100

prob_rhythm_only, prob_rhythm_and_standout, prob_smooth_and_standout, prob_smooth_or_standoutHere are the calculated percentages for the requested probabilities:

1. **Rhythm only:** 19.18%
2. **Rhythm and StandOut:** 12.33%
3. **Smooth and StandOut:** 5.48%
4. **Smooth or StandOut:** 45.21%

Would you like further details or clarification on any of these calculations?

Here are five related questions:
1. How would the probability change if more students participated in all three activities?
2. How can you calculate the probability of students participating in exactly two activities?
3. How is the probability of "Smooth only" different from the combined probabilities?
4. What is the complement of "Smooth or StandOut"?
5. How would you compute the probability of students participating in at least two activities?

**Tip:** In probability problems involving Venn diagrams, the total sum of probabilities for all distinct regions should always add up to 100%.

A survey was conducted at a local Halloween dance with 73 students asking if they had ever competed in three Halloween dance categories: Smooth, Rhythm, or StandOut. The results were summarized in a Venn diagram showing overlapping participation.

This problem covers a survey of 73 students about participation in three Halloween dance categories: Smooth, Rhythm, and StandOut. Using Venn diagrams and probability, learn how to calculate probabilities of different event combinations. Suitable for Grades 9-11.

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