Bayes’ theorem (alternatively Bayes’ law or Bayes' rule, also written as Bayes’s theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. With the Bayesian probability interpretation the theorem expresses how a subjective degree of belief should rationally change to account for availability of related evidence. Bayes' theorem is useful because it helps improve a weakness in our intuition about how probable it is a given hypothesis will occur. Bayes' theorem on Wikipedia
Below is an example from Daniel Kahneman's book Thinking Fast and Slow
A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. 85% of the cabs in the city are Green and 15% are Blue. A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colours 80% of the time and failed 20% of the time. What is the probability that the cab involved in the accident was Blue rather than Green knowing that this witness identified it as Blue?
(x*y) / (x*y) + (z*1 - z*x)
x = base rate
y = hit rate
z = false alarm rate
If we plug these variables into the calculator we see the probability the hit and run cab was blue is only 41%. Thus, Even though the witness is correct 80% of the time the actually probability the cab was blue is 41%. The theorem results are most influenced and weighted by the base rate (prior probability). The result is more likely to be near the base rate. This is true even if an evidence event seems to be high proof of the hypothesis. A base rate of 1% will result in 3.88% probability of hypothesis with a hit rate at 80%. If a base rate probability is less than 1%, even with a 90% hit rate probability, the result probability will only be 8.3%. It will take many new evidence events to even increase the hypothesis probability to greater than 50%.
If the result hypothesis is entered as the new base rate, and the other variables remain the same, one can get an intuition of how the probability of the hypothesis becomes more certain given the mounting evidence. The base rate must be as accurate, and relevant to the real life situation, as possible. If one doesn't know the hit rate, or false alarm rate, estimates can be supplied for these variables.