Solution:
The positioning of the needle relative to nearby lines can be described with a random vector which has components:
A [0, d )
[0, )
The random vector is uniformly distributed on the region [0,d)×[0, ). Accordingly, it has probability density function 1/d . The probability that the needle will cross one of the lines is given by the integral
p
0
l sin
0
1 d
2l dAd d