I'm getting $1/4$ as the probability in Buffon's needle problem and I'm having some trouble finding my mistake.
The problem statement:
If a needle of length 1 is dropped at random on a surface ruled
with parallel lines at distance 2 apart, what is the probability that the needle will cross one
of the lines?
I reduced the problem to a simpler one:
If a needle of length one is dropped in a region of length two such that its center lies on a horizontal line, creating an angle theta with that line, what is the probability that it will contact a vertical line at the center of said region?
(Visualization)
From simple trig rules, the needle will begin contact when the center is $\frac{\cos(\theta)}{2}$ away from the central line, and likewise ends contact at the same distance.
So we have a total sample space of 2 and a hit-space of $\cos(\theta)$, so we get a probability to hit of $\frac{\cos(\theta)}{2}$. (Assume $0\le\theta\le2\pi$)
Graphing our space over all possible theta we have:
Calculating the probability of placing a needle in the hit region of our sample space is simply the hit area (blue) divided by the total area.
So we get $$P=\frac{4(\frac{1}{2}\cdot1\cdot\frac{\pi}{2})}{4\pi}=\frac{1}{4}$$
We can extend these regions both horizontally and vertically without affecting the probability, so this 'solution' claims the answer to the problem is $1/4$.
What have I done wrong?
Best Answer
That isn't the graph of cosine!!!
The area under the curve of abs(cosx) between zero and 2pi is 4. The total sample space area is 4pi. So the probability is 1/pi as desired.