Is there a connection between Boolean algebra and probability? Is there a unifying abstraction that links Boolean algebra and probability theory? Both Boolean algebra and probability provide us the means to answer questions about set participation. On the one hand, Boolean algebra is an absolute, binary view of set participation -- either you're in or you're out (0 or 1). With probability we can think of expected "degrees" of participation within a set according to a real-valued probability that spans between 0 and 1. Is this about as far as I can draw a comparison? I'm interested to know if there has been work done to develop a deeper connection between a probability variable to a Boolean variable. Are there any sources that describe Boolean algebra as a limiting case of probability theory?

Take a look at “Probability Theory: The Logic of Science” by Jaynes. He casts probability as an extension of logic, developing it as Boolean algebra with uncertainty.