Probability of collisions - Computer networking problem
This is a problem in my text that I have the solution for. I do not understand how the solution is achieved however. I was hoping someone could enlighten me on the reasoning behind this. I took statistics nearly two years ago and it's faded some. :(
Problem: A disadvantage of a broadcast subnet is the capacity wasted when multiple hosts attempt to access the channel at the same time. As a simplistic example, suppose that time is divided into discrete slots, with each of the n hosts attempting to use the channel with probability p during each slot. What fraction of the slots will be wasted due to collisions?
Solution from text: Distinguish n + 2 events. Events 1 through n consist of the corresponding host successfully attempting to use the channel, i.e., without a collision. The probability of each of these events is:
Event n + 1 is an idle channel, with probability
Event n + 2 is a collision. Since these n + 2 events are exhaustive, their probabilities must sum to unity. The probability of a collision, which is equal to the fraction of slots wasted, is then just