DESCRIPTION

Students:  Please keep in mind the OMSI rules.  Save your files often,
make sure OMSI fills your entire screen at all times, etc.  Remember
that clicking CopyQtoA will copy the entire question box to the answer
box.

In questions involving code which will PARTIALLY be given to you in the
question specs, you may need add new lines.  There may not be
information given as to where the lines should be inserted.

If a question includes test code, make sure to include it in your
submission

Do not answer any question with simulation code unless this is specified. 

MAKE SURE TO RUN THE CODE IN PROBLEMS INVOLVING CODE!  Hit the OMSI
Submit and Run button.

QUESTION 

(Text answer.) In an adjacency matrix, the number of 1s in row i is
called the _______________________ of vertex i.

QUESTION -ext .R -run 'Rscript omsi_answer2.R'

(Code answer.) Consider the ALOHA example, p = 0.4 and q = 0.8,
but with three nodes, all of which start out active. The records at
terminal at node 1 show that there was a collision in epoch 1. (It does
not indicate whether that node was involved.)

Write simulation code to find the probability that all three nodes were
involved in that collision.

sim3coll <- function(nreps)
{


}

print(sim3coll(10000))

QUESTION -ext .R -run 'Rscript omsi_answer3.R'

(R code answer.)  Solve Problem 2 mathematically (i.e. "with mailing
tubes"), obtaining an exact answer.  

print(        )

QUESTION -ext .R -run 'Rscript omsi_answer4.R'

(R code answer.) Suppose m numbers are chosen at random, without
replacement, from 1,2,...,n.  Let X denote the length of the largest gap
between consecutive numbers in the chosen set.  The gap starting at 1,
and the one ending at n, do not count unless 1 or n is in the chosen
set, respectively.  For example, if n = 10 and m = 3, we might choose 2,
6 and 7.  The gaps would then be 4 and 1.  Write a function to find the
exact probability (this is not a simulation) that X = k. You must make
use of R's built-in functions combn() and diff().

maxgap <- function(n,m,k)
{


}

print(maxgap(6,3,3))  # 0.3
print(maxgap(8,4,3))  # about 0.39
print(maxgap(5,2,2))  # 0.3
print(maxgap(20,6,8))  # about 0.10