If $N = \sum_{i=0}^{n} I_i$ where $I_i$ is the indicator variable of event $A_i$, with $P(A_i)=p_i$, how can one show that
$$ P(N=0) \leq \frac{\operatorname{Var}(N)}{E(N)^2} $$
using Chebyshev's inequality or otherwise?
I don't know how to apply Chebyshev's inequality here – it doesn't seem like there's anything that has the right form.
Edit: The first part of the question is to calculate the expectation and variance of $N$ in terms of $p_i$ and $p_{ij} = P(A_i \cap A_i)$ - perhaps this could be useful for the second part?
Edit: Full question: