I want to prove the following, stated as an exercise is Mitzenmacher and Upfal's book:
Let $n$ points be placed uniformly at random on a circle of circumference 1. These points divide the circle into $n$ arcs, and we let $Z_i$, for $1 \leq i \leq n$ be the length of these circles. Then, for sufficiently large $n$, there exists a constant $c'$ such that at least one $Z_i$ is at most $c' / n^2$ with probability at least $1/2$.
There is a hint suggesting one use the second moment method, which should be the following lemma:
Let $X = \sum_{i=1}^n X_i$ where each $X_i$ is a 0-1 random variable. Then $Pr[X > 0] \geq \sum_{i=1}^n \frac{Pr[X_i = 1]}{E[X | X_i = 1]}$. Note that the $X_i$ need not be independent.
My idea is to take each $X_i$ to be the event "$Z_i$ is of length less than $c' / n^2$", which happens with $Pr[Z_i < c' / n^2] = 1 - (1 - \frac{c'}{n^2})^{n-1}$, which is also the expectation of $X_i$. By linearity of expectation the denominator of the above lemma should reduce to $\sum_{j=1}^n Pr[X_j = 1 | X_i = 1]$, which is $1$ for $j = i$, but I'm not sure what it is in the case of $j \neq i$.