Concentration inequalities (CIs) are a powerful tool that provide probability bounds on how a random variable deviates from its expectation. In this dissertation, first I describe a blockchain protocol that I have developed, called Graphene, which uses CIs to provide probabilistic guarantees on performance. Second, I analyze the extent to which CIs are robust when the assumptions they require are violated, using Reinforcement Learning (RL) as the domain. Graphene is a method for interactive set reconciliation among peers in blockchains and related distributed systems. Through the novel combination of a Bloom filter and an Invertible Bloom Lookup Table, Graphene uses a fraction of the network bandwidth used by deployed work for one- and two-way synchronization. It is a fast and implementation-independent algorithm that uses CIs for parameterizing an IBLT so that it is optimal in size for a given desired decode rate. I characterize performance improvements through analysis, detailed simulation, and deployment results for Bitcoin Cash, a prominent cryptocurrency. Implementations of Graphene, IBLTs, and the IBLT optimization algorithm are all open-source code. Second, I analyze the extent to which existing methods rely on accurate training data for a specific class of RL algorithms, known as Safe and Seldonian RL. Several Seldonian RL algorithms have a component called the safety test, which uses CIs to lower bound the performance of a new policy with training data collected from another policy. I introduce a new measure of security to quantify the susceptibility to corruptions in training data, and show that a couple of Seldonian RL methods are extremely sensitive to even a few data corruptions, completely breaking the probability bounds guaranteed by CIs. I then introduce a new algorithm, called Panacea, that is more robust against data corruptions, and demonstrate its usage in practice on some RL problems, including a grid-world and diabetes treatment simulation.