Sure, let's delve deeper into the problem and break down each part step-by-step.

### Step-by-Step Solution:

#### 1. Understand the given information:
- Total number of students: \( 40 \)
- Number of students enrolled in painting (\( P \)): \( 15 \)
- Number of students enrolled in both painting and crafts (\( P \cap A \)): \( 7 \)
- Number of students enrolled in crafts (\( A \)): \( 18 \)

#### 2. Determine the number of students enrolled only in crafts:
To find the number of students enrolled only in crafts, we subtract the number of students who are enrolled in both painting and crafts from the total number of students enrolled in crafts:

\[ \text{Number of students enrolled only in crafts} = A - (P \cap A) = 18 - 7 = 11 \]

#### 3. Calculate the probability:
The probability \( P(E) \) of an event \( E \) is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In this case, the event \( E \) is selecting a student who is enrolled only in crafts.

\[ P(\text{only in crafts}) = \frac{\text{Number of students enrolled only in crafts}}{\text{Total number of students}} = \frac{11}{40} \]

#### 4. Double-checking with Venn Diagram approach:
Visualizing the problem with a Venn diagram can help confirm our understanding:

- Draw two overlapping circles: one for painting (P) and one for crafts (A).
- The overlap (intersection) represents students enrolled in both painting and crafts.
- Place 7 in the intersection.
- Place \( 15 - 7 = 8 \) in the part of the painting circle that doesn't overlap with crafts (students only in painting).
- Place \( 18 - 7 = 11 \) in the part of the crafts circle that doesn't overlap with painting (students only in crafts).

So, the Venn diagram would show:
- 8 students only in painting.
- 7 students in both painting and crafts.
- 11 students only in crafts.

Adding these up:
\[ 8 + 7 + 11 = 26 \]

Since the total number of students is 40, the remaining \( 40 - 26 = 14 \) students are not enrolled in either painting or crafts.

### Final probability calculation:
The probability that a randomly chosen student is enrolled only in crafts is:

\[ P(\text{only in crafts}) = \frac{11}{40} \]

This can also be converted to a decimal or percentage if desired:
\[ P(\text{only in crafts}) = 0.275 \text{ or } 27.5\% \]

This detailed breakdown confirms that the probability a randomly chosen student is enrolled only in crafts is \( \frac{11}{40} \).

Would you like any further clarifications or more examples to solidify this concept? Here are 5 additional questions to consider:

1. What is the probability that a randomly chosen student is enrolled in painting?
2. How many students are enrolled only in painting?
3. What is the probability that a student is enrolled in both painting and crafts?
4. How many students are enrolled in neither painting nor crafts?
5. What is the probability that a student is enrolled in at least one of the courses?

**Tip:** Always check your work by considering the problem from multiple perspectives, such as using algebraic methods and visual aids like Venn diagrams.

Explore the detailed solution for calculating the probability that a randomly chosen student is enrolled only in crafts. This problem involves set theory concepts, specifically intersecting sets and probability calculations. Suitable for high school students studying probability and set theory.

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