What is Bayes' rule and why is it useful?

Bayes' rule is a really neat equation that's used everywhere, including in Artificial Intelligence! Most things in the real world aren't certain: that's why we use probabilities. Also, we often can't directly observe the quantity we want to know, and the method of observation can be imperfect: this is why we use Bayes' law. Imagine my doctor ran a test to tell me if there was something wrong with me. Without Bayes' law we wouldn't know the actual probability of there being something wrong with me because we wouldn't be able to account for errors in the test itself! What if the test was really bad and only got it right 10% of the time? That's what Bayes' law is for. To use Bayes' law to find the probability there's nothing wrong with me, we need two additional pieces of information that concern the test: the probability of the test returning a negative result (whether or not there's something wrong with the person being tested), and also the probability of the test being negative given there's nothing wrong with the person being tested. The equation is then P(A) = P(A|B)xP(B)/P(B|A). The most important part is making sure you understand what A and B mean (they are probabilistic events) and how to find the quantities on the right hand side if they aren't given to you. Running through some examples will help.