I think we are very familiar with probability being expressed in the ‘normal’ way: “the relative frequency of an event. If it occurs on average three out of four times, he or she will assign to it a probability of 3/4.” But to help understand the Bayesian way, think of probability as a way to give a belief a degree of certainty based on incomplete information; “all probability is conditional and subject to change when more data emerges.”
Once new information is available, we can use that prior knowledge we already had plus this new information to derive an updated assessment of the probability a conditional event will occur.
Reading this article from the Irish Times, Bayes’s Rule plays an imperative role. Calculating the probability of an event is a matter of life and death when it came to the 1983 launch of space shuttle Challenger which ended up exploding and killing all 7 crew members. According to the article, a Bayesian analyst estimated a 1 in 35 chance of a major accident occurring, but NASA’s estimate was 1 in 100,000 no doubt giving a false sense of security.