BUFFON’S NEEDLE & Jason Yu – 6 th THE MONTE CARLO 03/10/2012 METHOD
MONTE CARLO METHODThere are usually several characteristics of the Monte Carlomethod: 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.
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.
GEORGE-LOUISLECLERCThe pioneer oft h e B u f fo nNeedleex p e r i me n t .
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?”
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.
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:
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.