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:

$\displaystyle p(1-p)^{n-1}$

Event *n* + 1 is an idle channel, with probability

$\displaystyle (1-p)^n$

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

$\displaystyle 1-np(1-p)^{n-1}-(1-p)^n$.