Using Probabilistic Methods to Calculate Volumes

(due date December 6)

Consider the region, R, of 3 dimensional space consisting of all points (x,y,z) such that the sum of the distances from (x,y,z) to the four vertices of the unit tetrahedron is less than 4. So, as an example, the point (x,y,z)  shown in the accompanying diagram will belong to the region R if and only if the sum of the lengths of the dashed red lines will be less than 4. What is the volume of R?

One approach to answering this question would be to arrive at an algebraic, rather than geometric, description of the region R and then to calculate the associated volume integral by double integration. Since, to most people, this would seem a daunting task, this assignment is devoted to considering an alternative approach. Before continuing, use Maple's implicitplot3d fucntion to get a rough idea of the region R. In particular, you should be able to find a reasonably small box which entirely contains the region R. The volume of this box is, of course, easy to calculate; let us suppose that the volume of the box is V. What is the probability that a point chosen at random from this box will be cotnained in the region R? This probability shold be equal to the ratio of the volume of R to the volume of the box. In other words, if W denotes the volume of the region R, then th eprobability that a randomly chosen point belongs to R is W/V.

This observation can be used to approximate W, the volume of R. This is done by choosing many random points in the box enclosing R and keeping track of how often these points actually belong to R. After performing any such experiments you will have arrived at an approximation to the probability that a randomly chosen point belongs to R. For example, let us say that after 10 experiments you have found that 7 of the randomly chosen points belonged to R and the other 3 did not. You might conclude that the probability that a point chosen at random from the box belongs to R is 7/10. Hence 7/10 = W/V. Since you have already calculated V, it is now an easy matter to arrive at a value for W. Naturally, you would not expect 10 poitns to yield a very good approximation to the probability of belonging to R, so, the number of experiments you perform will have to be larger than 10.

But how does one perform these experiments? Maple has a function rand which, when invoked without any arguments, yields a random 12 digit integer. Hence typing rand() will yields a 12 digit integer but repeating the same instruction to maple will yield a different 12 digit integer. Dividing this output by 1,000,000,000,000 will yield a random numebr in the unit interval (up to 12 digit accuracy). Multiplying this by 3 and subtracting 5 will yeild a random number in the interval (-5, -2). Why? Using similar operations, it is possible to use rand() to obtain random numbers in any desired interval. However, this does not immediately solve theproblem of finding random points in some given box because points are not numbers but triples of numbers. If the box is a product of interval [a,b] ? [c,d]?[e,f] then rand() can be used three separate times to find random numbers in the intervals [a,b], [c,d] and [e,f]. These can be used to determine a random point in the box.

The final task which Maple must perform is to determine whether a given a random point chosen from the box belongs to the region R. This can be done with Maple's if ... then.. fi structure. Repeating this operation several times will yield an approximation to the probability desired.

In your conclusion you should discuss the accuracy of your results. There are theoretical ways of determining the accuracy of an approximation such as the one being considered, but these will be left to a course in statistics. A less sophisticated way of gaining some idea of the accuracy of your estimation of the probability that a point belongs to R is to repeat your experiment with the same number of points and see how much the new answer differs from the old. Doing this a third time will give you even more information.


Instructor

Mike Zabrocki
Email address: zabrocki@yorku.ca
Department of Mathematics and Statistics
York University
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