Parallel lines are drawn on a playground such that D' is the distance maintained between these lines, using white Chalk.

A stick of length 'L' (L<=D) is thrown on the ground. Determine the probability that the stick intersects a line.

This problem is a classic example of **Buffon's Needle problem**, a famous problem in probability theory. The goal is to find the probability that a stick of length \( L \) (where \( L \leq D' \), the distance between two parallel lines) intersects one of the parallel lines when randomly thrown on the ground.

### Explanation:
Let’s break it down step by step:

1. **Setup**:
   - There are parallel lines spaced \( D' \) units apart.
   - A stick (needle) of length \( L \) is thrown randomly onto the surface.

2. **Condition for Intersection**:
   For the stick to intersect a line, two factors matter:
   - The distance from the center of the stick to the nearest line, which we will call \( x \). It can range from 0 to \( \frac{D'}{2} \) (the maximum possible distance from the center of the stick to a line).
   - The angle \( \theta \) that the stick makes with respect to the lines. This angle can range from 0 to \( \pi/2 \) (since the stick’s orientation is random, and only angles up to \( \frac{\pi}{2} \) matter due to symmetry).

   The stick will intersect a line if the distance from the center of the stick to the line is less than or equal to half the projection of the stick onto the perpendicular of the line:
   \[
   x \leq \frac{L}{2} \sin \theta
   \]
   where \( \sin \theta \) gives the vertical projection of the stick onto the line.

3. **Probability Calculation**:
   The probability of intersection is calculated by integrating over all possible positions \( x \) of the center of the stick and all possible angles \( \theta \). The probability of an intersection, \( P \), is given by:
   \[
   P = \frac{2L}{\pi D'}
   \]

   This formula assumes that \( L \leq D' \). If the stick is longer than the distance between the lines, the formula would need adjustment.

### Final Result:
The probability that the stick intersects a line is:
\[
P = \frac{2L}{\pi D'}
\]

This formula shows that the probability depends on the ratio of the stick's length \( L \) to the distance between the lines \( D' \).

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Would you like further details on any of the steps, or do you have questions? Here are 5 related questions to consider:

1. How would the probability change if the length of the stick exceeds the distance between the lines?
2. What is Buffon's Needle problem in more general cases?
3. Can you derive the formula for \( P \) using integration?
4. What happens if the lines are not equally spaced?
5. How does the formula change if the stick is thrown at an angle to the lines?

**Tip**: Understanding how to calculate probabilities in geometric settings often relies on symmetries and averages over random variables like angles or distances.

Parallel lines are drawn on a playground such that D' is the distance maintained between these lines, using white chalk. A stick of length 'L' (L <= D) is thrown on the ground. Determine the probability that the stick intersects a line.

Learn how to calculate the probability that a stick of length L intersects parallel lines spaced D' units apart using Buffon's Needle formula. This geometric probability problem is a classic in probability theory, perfect for advanced high school students in Grades 11-12.

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