The Monte Carlo method involves generating random inputs from a probability distribution, performing deterministic computations on those inputs, and aggregating the results. An early example is Buffon's needle experiment, which aimed to estimate pi by dropping needles onto a floor divided into parallel strips. The probability a needle crosses a strip is related to pi, allowing pi to be estimated through repetition of the experiment. While accurate estimates of pi were obtained this way, the experiment is prone to manipulation by continuing trials until a desired precision is reached.

1. BUFFON’S NEEDLE &
Jason Yu – 6 th
THE MONTE CARLO 03/10/2012
METHOD

2. MONTE CARLO METHOD
There are usually several characteristics of the Monte Carlo
method:
 Define a domain of possible inputs.
 Generate inputs randomly from probability distribution over the
domain.
 Perform a deterministic computation on the inputs.
 Aggregate the results.
An early version of the Monte Carlo method can be seen in the Buffon’s
Needle Experiment.

3. MONTE CARLO METHOD – WHY PI?
 The Monte Carlo method is heavily intertwined with the
process of estimating pi.
 Let’s consider a circle inscribed in a unit square. If the circle
and square have rations of areas that is pi/ 4, the value of pi
can be approximated by using steps in the Monte Carlo
method:
 Draw a square on the ground, then inscribe a circle within it.
 Uniformly scatter some objects of uniform size over the square.
 Count the number of objects inside the circle and the total # of
objects.
 Lastly, the ration of the two counts will be an estimate of the ratio of
the two areas, which is pi/4. Multiply the result by 4 and you should
receive an estimate for pi.

4. GEORGE-
LOUIS
LECLERC
The pioneer of
t h e B u f fo n
Needle
ex p e r i me n t .

5. BUFFON’S NEEDLE – THE QUESTION
 In the 18 th Century, a French naturalist and renowned
mathematician, George -Louis Leclerc, the Comte de
Buf fon, proposed a question that states:
“Suppose we have a floor made of parallel strips of wood, each the
same width, and we drop a needle onto the floor. What is the
probability that the needle will lie across a line between two strips?”

6. BUFFON’S NEEDLE – THE EXPERIMENT
 Buf fon’s needle, the earliest problem in geometric probability
to be solved, can be solved using integral geometry.
 In the experiment, we are trying to find probability, which can
be rearranged as
 So if we can derive an equation to find probability, we can
likewise determine a rough estimate for pi.

7. BUFFON’S NEEDLE – THE MATH
 If we drop n needles and find that h of those needles are
crossing lines, we can determine that P is approximated by
the fraction h/n.
 Therefore, we can derive the formula:

8. BUFFON’S NEEDLE – WHY PI?
 In 1901 Italian mathematician Mario Lazzarini performed the
Buffon’s needle experiment and concluded, after tossing a
needle 3408 times, that the estimate for pi was 355/113.
 This value is extremely accurate and differs from the actual
value of pi by no more than 3E-7.
 However, there is some controversy surrounding this experiment
as it would be rather easy to manipulate the results by simply
repeating the process.
 For example, if one drops 213 needles and happens to get 113
successes, then one can report an estimate of pi accurate to six
decimal places. However, if this doesn’t work, one can still
perform 213 more trials and hope for 226 successes.