https://file.aiquickdraw.com/mathbot/file/1728259729062fqpcus88.jpg

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.

A survey was conducted at a local ballroom dance studio asking 72 students if they had ever competed in the following dance categories: Smooth, Rhythm, or Standard. The results were presented in the Venn Diagram. A single dancer is selected at random. What is the probability that: a) The dancer has competed in the Smooth category. b) The dancer has competed in the Rhythm and Standard categories. c) The dancer has competed in the Smooth or Standard categories. d) The dancer has competed in the Standard category, given that they have competed in the Rhythm category.

Explore probability questions based on a Venn diagram showing the results of a survey of 72 ballroom dancers who competed in Smooth, Rhythm, or Standard categories. Includes key concepts like probability and conditional probability, with detailed explanations for solving each question.

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