Buffon's Needle: A Surprising Probability Involving π

A French nobleman, Georges Louis Leclerc, Comte de Buffon, posed the following problem in $1777$:

“Suppose that you drop a short needle on ruled paper-what is then the probability that the needle comes to lie in a position where it crosses of the lines?” 

The probability depends on the distance $d$ between the lines of the ruled paper, and it depends on the length $\ell$ of the needle that we drop-or rather it depends only on the ratio $A$ short needle for our purpose is one of length $l\le d$. In other words, a short needle is one that cannot cross two lines at the same time (and will come to touch two lines only with probability zero). 

The answer to Buffon’s problem may come as a surprise: It involves the number $\pi$.