WHAT IS BUFFON'S NEEDLE?

BUFFON'S NEEDLE PROBLEM:

what is Buffon's needle problem?
It is interesting, it is the problem of calculating the approximate value of pi.

what's the problem is?
Throw a needle of length L at random on a floor covered by equidistant parallel lines d units apart. What is the probability that the needle will cross at least one of the lines?

SOLUTION:

Let the length of the needle is one unit and the distance between the lines is also one unit. There are two variables, the angle at which the needle falls (theta) and the distance from the center of the needle to the closest line (D). Theta can vary from 0 to 180 degrees and is measured against a line parallel to the lines on the paper. The distance from the center to the closest line can never be more that half the distance between the lines.
The needle will hit the line if the closest distance to a line (D) is less than or equal to 1/2 times the sine of theta. That is, D <= (1/2)sin(theta).
we now plot D along the ordinate and (1/2)sine(theta) along the abscissa. The values on or below the curve represent a hit (D <= (1/2)sin(theta)). Thus, the probability of a success it the ratio shaded area to the entire rectangle.
The shaded portion is found with using the definite integral of (1/2)sin(theta) evaluated from zero to pi. The result is that the shaded portion has a value of 1. The value of the entire rectangle is (1/2)(pi) or pi/2. So, the probability of a hit is 1/(pi/2) or 2/pi. That's approximately .6366197.

To calculate Pi from the needle drops, simply take the number of tries and multiply it by two, then divide by the number of hits, or

2(total tries) / (number of hits) = pi (approximately).