A random interval graph of order n is generated by picking 2n numbers X1,…,X2n independently from the uniform distribution on [0,1] and considering the collection of n intervals with endpoints X2i−1 and X2i for i ∈ {1,…,n}. The graph vertices correspond to intervals. Two vertices are connected if the corresponding intervals intersect. This paper characterizes the fluctuations of the independence number in random interval graphs. This characterization is obtained through the analysis of the greedy algorithm. We actually prove limit theorems (central limit theorem and large deviation principle) on the number of phases of this greedy algorithm. The proof relies on the analysis of first-passage times through a random level.