Bayes' theorem, sometimes called Bayes' rule or the principle of inverse probability, is a mathematical theorem that follows very quickly from the axioms of probability theory. In practice, it is used to calculate the updated probability of some target phenomenon or hypothesis H given new empirical data X and some background information, or prior probability.
The prior probability of some hypothesis is usually represented by some percentage between 0% and 100%, or some number between 0 and 1. This probability is often called degree of confidence, and is meant to vary from observer to observer, as not all observers have had the same experience and therefore cannot make equivalent probability estimates for any given hypothesis. The application of Bayes' theorem in a scientific context is called Bayesian inference, which is a quantitative formalization of the scientific method. It allows the optimal revision of theoretical probability distributions given experimental results.
Bayes' theorem in the context of scientific inference says the following: "The new probability of some hypothesis H being true (called posterior probability) given new evidence X is equal to the probability that we would observe this evidence X given that H is actually true (called conditional probability, or likelihood), times the prior probability of H being true, all divided by the probability of X."
A common restatement of the above in terms of how a test result contributes to the probability that a given patient has cancer can be shown as the following:
p(positive|cancer)*p(cancer) + p(positive|~cancer)*p(~cancer)
The vertical bar means "given." The probability the patient has cancer after a positive result on a certain cancer test is equivalent to the probability of a positive result given cancer (derived from past results) times the prior probability of any given person having cancer (relatively low) all divided by that same number, plus the probability of a false positive times the prior probability of not having cancer.
It sounds complicated, but the above equation can be used to determine the updated probability of any hypothesis given any quantifiable experimental result.