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.