Icot
This is Buffon’s needle problem that uses integral geometry to approximate pi if the length of the pencil is not greater than the width of the line.
let
L= length of pencil
t= parallel lines are t units apart
x= the distance from the center of the pencil to the closest line
?= the acute angle between the pencil and the lines
The probability density function of x between 0 and t /2 is
(2/t)dx.
The probability density function of ? between 0 and ?/2 is
(2/?)d?.
x and ? are independent random variables, so the joint probability density function is
(4/t?)dxd?.
The pencil crosses the line when
x<=(L/2)sin?.
1. If it’s a short pencil & L<=t:
Integrate the joint probability density function to get the probability that the pencil will cross a line:
= 2L/t?
2. If it’s a longer pencil, the calcualtions r more complicated but solvable