Let's say that the centre point of the needle falls between two of the lines, and the angle between the needle and the lines is $\theta$ . Given a certain value of $\theta$ , where does the needle need to fall in order to cross one of the lines?

We can see that the needle will cross a line if its centre is within $\frac12\sin\theta$ cm of one of the lines. Then, the probability that a needle at this angle will lie across a line is $\frac{2\left(\frac12\sin\theta\ \text{cm}\right)}{2\text{cm}}=\frac12\sin\theta$ Then, integrating over possible values of $\theta$ , the probability that any needle will cross a line is $\frac2{\pi}\int_0^{\frac{\pi}2} \frac12\sin\theta\ d\theta=\frac1{\pi}\left[-\cos\left(\frac{\pi}2\right)+\cos(0)\right]=\frac1{\pi}$ This gives the ratio of needles lying across lines to total needles. We want the ratio the other way round, so the answer is $\frac1{1/\pi}=\pi$ .

I've written a Python Program to figure it out.

Buffon needle

* CODE STARTS *

import random, math

p = 100

total = 20005 touch = 0

for i in range (0, total):

y = random.randrange(0*p, 6*p)/p # Dist. y from below
a = round((random.randrange(0, 360*p)/p)*math.pi/180, 5) #Angle (Rad)

h = 0.5*math.sin(a)
up = y+h
do = y-h

if up < 0:
    up = int(up)-1
else:
    up = int(up)

if do < 0:
    do = int(do)-1
else:
    do = int(do)

if up < do:
    up, do = do, up

if ((do == 0 and up == 1) or (do == 2 and up == 3) or (do == 4 and up == 5) or (do == 6 and up == 7)):
    touch += 1

print(total/touch)

* CODE ENDS*

==> 3.143463230672533