I want to solve Buffon's Needle problem but first I was trying to tackle a simpler case.
So: consider an infinite line with points each $t$ units. Let's say that we have a "needle" of length $L\leq t$ which we throw. I claim the probability of the needle not touching any point is given by:
$$ P = \frac{t-L}{t} = 1 -\frac{L}{t} $$
That's because we have a length of size $t-L$ where the needle can lie and the total length between points is $t$
Now, in the 2D case we simply consider $P_\theta = 1 - \frac{L\sin(\theta)}{t}$ (i.e., the projection of the needle's length along the y axis) and sum (integrate) from $0$ to $\pi$.
However the result differs from the usual $\left(\frac{2L}{\pi t}\right)$. Am I missing something in the integration? Or maybe the reasoning from the 1D case is incorrect?
Thanks for your advise