TITLE: Lower bound on Tail Probabilities
QUESTION [8 upvotes]: Inequalities such as Markov's and Chebyshev’s provide upper bounds on tail probabilities. Are there similar inequalities that give lower bounds in the form $P(X \geq \alpha)>\theta$?

REPLY [6 votes]: Markov's inequality is also called the first moment method. What you want is the second moment method using bounds for the two first moments to derive the desired inequality :
http://en.wikipedia.org/wiki/Second_moment_method