Notation: Prob(A) means "the probability of event A" and Prob(A|B) is "the probability of event A, given that event B has happened."
Bayes' Theorem: Prob(A|B)xProb(B) = Prob(B|A)xProb(A)
Now, Prob(A|B) and Prob(B|A) are often confused by even the most intelligent of people. The confusion often appears in legal cases and is sometimes called the Prosecutor's Fallacy. Bayes' Theorem relates these two distinct conditional probabilities.
Followed by a straightforward example of why this really matters.