Boole is well known for so-called Boolean algebra. However, his almost totally forgotten work in how the mind works, logic, and probability and statistics, his "An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities", is worth a read.
I wanted to show an example of his thinking in the "simplest" case. Consider
Note that a is the a priori probability of the hypothesis, and p is the probability of y given x is true.
Putting this all together,
P = prob(xy)/prob(y) = ap/prob(y), and is the a posteriori probability of the physical hypothesis.
This can be simplified to
P = ap/(ap + c(1-a))
Note that the term c is the probability that if x does not occur, y occurs.
Please see this spreadsheet for an example where you can play with the numbers and see how the probabilities change.
Please check out the following books
Thanks for reading!
If you enjoyed any of my content, please consider supporting it in a variety of ways: