Here is a cool problem that I had in one of the problem sets for my Probability and Random Processes course.

Let $(X_1, X_2, \ldots, X_n)$ be independent random variables such that

$& \Pr(X_i = 1 - p_i) = p_i $&

and

$& \Pr(X_i = -p_i) = 1 - p_i $&

Let $(X = \sum_{i=1}^{n} X_i)$. Prove $& \Pr(|X| \geq a) \leq 2e^{-2a^2/n} $&.

Hint: You may use the following inequality $&\pi e^{\lambda(1-\pi)} + (1-\pi)e^{-\lambda\pi} \leq e^{\lambda^2/8}$&