Abstract
It's obvious that the largest possible value of a random quantity is at least as large as the mean value of that quantity, and that the smallest possible value is at most as large as the mean value. Surprisingly, this very basic observation turns out to be extremely useful if the random quantity is well chosen. For example, by examining the average number of cliques and stables of a given size in a random graph, Erdős deduced that there are graphs for which the largest clique and the largest stable are both very small. This proof gave rise to the probabilistic method in Combinatorics. By using the variance of random quantities, we can take the method a step further and prove results that do not follow from consideration of the mean alone.
