Homework 1
Due Monday, October 6.

This assignment will serve as a review of probability theory, and an introduction to its use in computer network analysis. (If you are curious, click here for a typical example of how deeply probabilistic analysis is used at the graduate study and research level. Of course, our usage in this undergraduate course will be much less and much simpler than the level of that research paper.)

Suppose we have three network nodes, each connected to two transmission lines, named line 1 and line 2. So, there are two wires emanating from each node to the lines.

In any given time slot, a node will have a message to send with probability p, in which case it will choose one of the two lines at random and attempt to send on that line. If two or more nodes attempt to send on the same line, a "collision" will occur, and none of the colliding messages will be received intact.

Find the following two quantities as functions of p:

(a) P(a message is sent intact on line 1 and no transmissions are attempted on line 2).

(b) P(line 1 has a collision | line 2 has no collision).

These problems may look difficult at first, but actually they are easy. It is simply a matter of enumerating all the possibilities.

As a check on your work, the answer to (a) in the special case of p = 0.1 is 0.1215, and the answer to (b) in that case is 0.0073.