The beta distribution: In the free book "think stats Probability and Statistics for Programmers" there is a chapter about how the beta distribution can be used as a prior to model an unknown probability and how Bayes' Theorem allow us to update that prior with a posterior distribution that is also a Beta distribution, that important property is called the self-prior property of the beta. Since the two parameters of the beta in that intuitive explanation are just the number of experiments (battings) and the number of successes (runs) that example constitutes a very intuitive and clear way to explain what is Bayesian Statistics. I think that you would enjoy the think stats book, it is aimed for programmers and it tries hard to enhance intuition.
I also enjoyed how you describe the atmosphere in your office, it seems that you work in a lovely place in which statistics is a well respected tool and people try to explore and innovate in a fun way without excessive pain. Nice post, I enjoyed it.