Weirdly: adv. In a strikingly odd or unusual manner
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14-04-2007 A good math puzzle

This blurb is about a puzzle that was presented to a math class I attended in college many years ago. I thought about it again recently, and think I have the answer. Sadly, I can't find the professor who presented it, so confirming it is tricky. I think I have it right though, so I present it here, and if anyone can show me that yes, I'm right or no, I'm way off, I'd be quite grateful.

The puzzle is this:

Imagine a plane (no, not the flying kind, but a surface), that is marked by a continuing series parallel lines. For the sake of this question, you can imagine these lines going on infinitely, and there being an infinite number of them (covering our infinite plane). These lines are all exactly the same distance apart. Now imagine a straight stick (or in this case, a line segment) is thrown on to the plane. The length of this line segment is exactly the same as the distance between the lines. It is dropped at a random angle in a random location. What is the probability of it intersecting one of the lines on the field?

I've thought about this a few different ways, but only one really seemes to make sense. This is the answer I've settled on:

For ease of comprehension, let's envision all of our field lines as horizontal ones. In truth it doesn't really matter, as the angle of them depends on our frame of reference. In this case, we'll say they all have a slope of zero.

To consider the overall probablillity of it hitting a line, we first have to consider the probablillity of it hitting a line when it lands at any given angle. That's actually fairly simple to see, although I have no idea how to prove it mathematically. First, keep in mind that the stick has two dimensions, Δx and Δy. As our lines go on indefinitely, and there is nothing to collide with except those lines, we can say that the horizontal range of our stick (Δx), does not matter. What does matter is the vertical range of our stick, Δy.

By observation, we can see that the chance of our stick crossing a line will be equal to its vertical length (Δy), divided by the actual distance between the lines on the field.

So what is that vertical length? Well, with a little bit of basic trig, we can see that it will be the length of our line segment multiplied by the sine of its angle relative to the lines.

This means that the probability (hereafter called "P") of the segment intersecting a line will be equal to the sine of our angle (we'll call it "a") multiplied by the length of our line (which we'll call "L"), divided by the distance between the lines (which is also equal to L). As a result, we get:

	P = sin(a) × L ÷ L
	P = sin(a)
So the chance of it crossing a line at any given angle is equal to the sine of that angle.

With that in mind then, we know the chance of it hitting a line if it's dropped at a given angle in a random location. What we want to know though, is what chance it has of hitting a line if the angle is random as well. To start, let's consider a graph of our probability over potential landing angles. The total range of our probability would be from 0 to 1. With the angle, (in radians), we only need to consider the range between 0 and π. This is because the range from π to 2π gives us an identical set of numbers and we are not interested in the numbers themselves, but the ratio between them - which remains the same. The graph would look like this:

Graph of probability over angle

This graph shows us the probability of the stick hitting a line, depending on the angle it falls at. In that case, its overall chance of hitting a line would be the area betwen this curve, "y = sin(x)" and the base of our graph, "y = 0", divided by the total area of the graph.

So, to calculate that we would take the integral of y = sin(x), for the range 0 to π, which gives us the area under the curve, and divide it by π, which is the total area of the graph.

The integral of the sine function is the negative cosine, so we get:

p  =  
sin(x) dx   =  
-cos(π) - (-cos(0))
cos(0) - cos(π)
1 + 1
  ≈   0.6366

So overall, our chance of intersecting a line ends up at approximately 63.7%

The catch here is, I don't honestly know if I'm right. I could be way off. I think I've got it right though, so if anyone would like to confirm it one way or another, I'd be glad to hear some feedback.