We derive two conditional expectation bounds, which we use to simplify cryptographic security proofs. The first bound relates the expectation of a bounded random variable and the average of its conditional expectations with respect to a set of i.i.d. random objects. It shows, under certain conditions, that the conditional expectation average has a small tail probability when the expectation of the random variable is sufficiently large. It is used to simplify the proof that the existence of weakly one-way functions implies the existence of strongly one-way functions. The second bound relaxes the independence requirement on the random objects to give a result that has applications to expander graph constructions in cryptography. It is used to simplify the proof that there is a security preserving reduction from weakly one-way functions to strongly one-way functions. To satisfy the hypothesis for this bound, we prove a hitting property for directed graphs that are expander-permutation hybrids.