| This thesis is divided into two areas of combinatorial probability: probabilistic divide-and-conquer, and random Bernoulli matrices via novel integer partitions.;Probabilistic divide-and-conquer is a new method of exact sampling that simulates from a set of objects by dividing each object into two disjoint parts, and pieces them together.;The study of random Bernoulli matrices is driven by the asymptotics of the probability that a random matrix whose entries are independent, identically distributed Bernoulli random variables with parameter 1/2 is singular. Our approach is an inclusion-exclusion expansion for this probability, defining a necessary and sufficient class of integer partitions as an index set to characterize all of the singularities. |
