A floor consists of parallel planks which have a two foot width. A circular disc of
diameter one foot is randomly and uniformly dropped onto the planks. Determine the
probability that the disc does not touch the line between two planks. Determine also
the expected distance from the center of the disc to the closest line between two planks
I think that you have given the distance between the two planks to be 2 feet, but I am not quite sure on this one. I think the technique would be to determine the surface area of that disc that would come in contact with the plank upon reaching the floor. I am quite intrigued by this problem, as I thought you would need more information to solve it, but I would like to wait and see how it can be solved.
Take a look at any web site discussing Buffon's Needle
Here is an actual calculation of the problem.
Yours involves 2 feet gaps, he works at 2 inches per gap.
So it will be the same.
http://www.youtube.com/watch?v=Vws1jvMbs64
To determine the probability that the disc does not touch the line between two planks, we need to calculate the area where the disc can land without touching the line.
To begin, let's visualize the situation. We have parallel planks with a width of two feet. This means there is a two-foot space between each pair of adjacent planks. A circular disc with a diameter of one foot is randomly dropped onto the planks.
The key observation here is that if the center of the disc falls within one foot on either side of the plank line, it will touch the line. Therefore, we need to determine the area where the center of the disc cannot fall.
Imagine drawing a one-foot boundary on either side of the plank line. This would create a three-foot space between the two boundaries, where the center of the disc cannot fall. Since each plank is two feet wide, this three-foot space consists of two one-foot boundaries and one two-foot plank.
The probability that the disc does not touch the line between two planks is equivalent to the ratio of the area where the disc can safely fall to the total area of the planks.
Let's calculate this probability step by step:
1. Calculate the total area of the planks:
- Assuming the length of each plank is L, the total area is L times the width of the planks.
- Let's assume there are N planks. So the total area of the planks is L * N.
2. Calculate the area where the disc cannot fall:
- Since the disc cannot fall within the three-foot space between the boundaries, the area is 3 * L.
3. Calculate the area where the disc can safely fall:
- Subtract the area where the disc cannot fall from the total area of the planks.
- Safe area = Total area of the planks - Area where the disc cannot fall.
4. Calculate the probability:
- Probability = Safe area / Total area of the planks.
To find the expected distance from the center of the disc to the closest line between two planks, we need to consider the possible positions where the center can fall and calculate the average or expected distance.
The closest line to the center of the disc can be either the left boundary or the right boundary. Since the disc is dropped randomly and uniformly, the chances of it falling closer to one boundary compared to the other are equal.
We need to calculate the distance from the center to the left boundary (let's call it d) and the distance from the center to the right boundary (also d). The expected distance will be the average of these two distances.
Let's denote the length of each plank as L (in feet). Here's how to calculate the expected distance:
1. Find the distance from the center to the left boundary:
- d = (L / 2) - 0.5 (since the left boundary is 0.5 feet away from the center)
2. Find the distance from the center to the right boundary:
- d = (L / 2) - 0.5 (since the right boundary is 0.5 feet away from the center)
3. Calculate the expected distance by taking the average of the distances to the two boundaries:
- Expected distance = (distance to the left boundary + distance to the right boundary) / 2.
That's how you can determine the probability that the disc does not touch the line between two planks and calculate the expected distance from the center of the disc to the closest line between two planks.