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Mathematical Puzzles
Needle drop problem

Sun Jul 25, 2010 1:21 am  by swedenman

Let's say we have an infinitely large floor that has horizontal lines that are perfectly parallel and exactly 1 inch apart. If a needle of length 1 is thrown at random on the floor, what is the probability it will intersect a line?

Hint: This question requires calculus. If you don't know calculus, you probably shouldn't waste your time on this.
Tue Aug 31, 2010 8:25 pm  by Excursion

Is it Pi? (3.1416etc) 8)
Thu Oct 07, 2010 3:43 am  by Pseudobyte

Let's say our pin lands such that its center is a distance x from the nearest line, where x goes from 0 to 1/2". The pin is at an angle θ, where 0 means the pin is perpendicular to the lines and π/2 means the pin is parallel to the lines. Without loss of generality, we can consider θ only in the range 0 to π/2.

The pin forms a right triangle, where the hypotenuse is 1/2", and the adjacent side is 1/2" cos(θ). The adjacent side tells us how far towards the nearest line the pin reaches or the maximum x the pin can land at to intersect the line at a given θ. If we divide this by our maximum x, 1/2", that gives us cos(θ), the probability that a pin at an angle θ will land at an x that causes it to intersect a line.

Integrating cos(θ) from 0 to π/2 and dividing by (π/2 - 0) gives the average probability for a pin to intersect a line across all theta. That integral gives us sin(θ) evaluated from 0 to π/2, or sin(π/2) - sin(0), which is 1. Dividing this by the size of the range, π/2, gives us 2/π, which is about 63.66%.

Interesting problem. I first tried doing x first and then θ, which gave me an arccosine I didn't have the faintest idea how to integrate. Then I realized if I did it in the other order, I'd just get a regular old cosine, giving me the solution above.

Of course now that I've looked up the integral of arccos and found out that it's doable using integration by parts, I feel silly. Calc 2 was my favorite calc class too. :'(
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