Finding Areas Using the Monte Carlo Method

The Monte Carlo Method gets its name from the city of Monte Carlo and the games of chance that are played in the casinos there. In mathematics this name is used whenever a problem is solved by a method that uses random numbers. The Monte Carlo method has been used in the following:
  • in virtually every recreational computer game to decide what the antagonists in the game will do next,
  • in computer simulations of processes that involve some element of randomness, such as the diffusion of neutrons out of a nuclear reactor or customers arriving at a queue,
  • in estimating the area of complicated objects.


Finding Areas: It is very difficult to use calculus to find the area of an object such as the one shown to the right. But using the Monte Carlo Method it's easy.
Here are the steps:
  • Put the object inside a rectangle of known area. Suppose that this rectangle has an area of 50 cm2.

  • Place a known number of points, say 100, at random locations inside the rectangle.

  • Count the number of random points that lie inside the object.

  • The area of the object is proportional to the number of points that lie inside it and is given by this formula:

If you count you will find that 22 points lie inside the object. Thus our extimate of the area is:


If we add another 900 points at random inside the rectangle (for a total of 1000) we get an improved estimate of the area. We now find that 280 points lie inside the object. This puts the area at:



Two of the advantages of the Monte Carlo method over other area-finding methods are that the accuracy improves with each random point that is added and that it can more easily be generalized to multi-dimensional integrals. Its biggest disadvantage is its slow convergence with increased numbers of points.

You may have wondered if there is a way to get the computer to count the points inside the object for you. The answer is yes. This is called the Inside or Outside Problem.
 

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