Needle problem in Probability A grid of lines distance $a$ apart and another grid of lines distance $b$ apart are perpendicular, dividing plane into rectangles. A needle of length $l$=min$\{a,b\}$ is thrown at random onto the grid. What is the probability it intersects lines in both grids simultaneously?

I have no idea how to approach such a problem. Any help would be appreciated.
Assume that the vertical lines are $a$ apart and the horizontal lines $b$ apart, $\ell=a\leq b$. The needle $N$ has a random inclination $\phi$ with repect to the horizontal, whereby we may assume that $\phi$ is uniformly distributed in $\bigl[0,{\pi\over2}]$, making use of symmetry. Given $\phi$, the probability that $N$ intersects a vertical line is ${a\cos\phi\over a}$, 
 the probability that $N$ intersects a horizontal line is ${a\sin\phi\over b}$, and these two events are independent. It follows that the overall probability $P$ that $N$ intersects lines of both kinds is given by
$$P={2\over\pi}\int_0^{\pi\over2}\&gt;{a\cos\phi\over a}{a\sin\phi\over b}\&gt;d\phi={a\over\pi b}\ .$$
Hint. For one set of parallel lines this is Buffon's Needle Problem. There are many solutions on the internet.

That should get you started dealing with two sets of parallel lines.

https://en.wikipedia.org/wiki/Buffon%27s_needle

http://www.cs.umb.edu/~eb/piday/
http://www.cs.umb.edu/~eb/piday/whypi.pdf