bayes theorem named after 18th century british mathematician thomas bayes is a mathematical formula for determining conditional probability conditional probability is the likelihood of an outcome occurring based on a previous outcome in similar circumstances bayes theorem provides a way to revise existing predictions or theories update probabilities given new or additional evidence 
in finance bayes theorem can be used to rate the risk of lending money to potential borrowers the theorem is also called bayes rule or bayes law and is the foundation of the field of bayesian statistics investopedia lara antalunderstanding bayes theoremapplications of bayes theorem are widespread and not limited to the financial realm for example bayes theorem can be used to determine the accuracy of medical test results by taking into consideration how likely any given person is to have a disease and the general accuracy of the test bayes theorem relies on incorporating prior probability distributions in order to generate posterior probabilities in bayesian statistical inference prior probability is the probability of an event occurring before new data is collected in other words it represents the best rational assessment of the probability of a particular outcome based on current knowledge before an experiment is performed posterior probability is the revised probability of an event occurring after considering the new information posterior probability is calculated by updating the prior probability using bayes theorem in statistical terms the posterior probability is the probability of event a occurring given that event b has occurred special considerationsbayes theorem thus gives the probability of an event based on new information that is or may be related to that event the formula also can be used to determine how the probability of an event occurring may be affected by hypothetical new information supposing the new information will turn out to be true for instance consider drawing a single card from a complete deck of 52 cards there are four kings in the deck so the probability that the card is a king is four divided by 52 which equals 1 13 or approximately 7 69 now suppose it is revealed that the selected card is a face card the probability the selected card is a king given it is a face card is four divided by 12 or approximately 33 3 as there are 12 face cards in a deck formula for bayes theoremexamples of bayes theorembelow are two examples of bayes theorem the first example shows how the formula can be derived from a stock investing example using amazon com inc amzn the second example applies bayes theorem to pharmaceutical drug testing bayes theorem follows simply from the axioms of conditional probability which is the probability of an event given that another event occurred for example a simple probability question may ask what is the probability of amazon com s stock price falling conditional probability takes this question a step further by asking what is the probability of the amzn stock price falling given that the dow jones industrial average djia index fell earlier the conditional probability of a given that b has happened can be expressed as if a is amzn price falls then p amzn is the probability that amzn falls and b is the djia is already down and p djia is the probability that the djia fell then the conditional probability expression reads as the probability that amzn drops given a djia decline is equal to the probability that amzn price declines and djia declines over the probability of a decrease in the djia index p amzn and djia is the probability of both a and b occurring this is also the same as the probability of a occurring multiplied by the probability that b occurs given that a occurs expressed as p amzn x p djia amzn the fact that these two expressions are equal leads to bayes theorem which is written as p amzn and p djia are the probabilities that amazon and the dow have fallen independently of each other the formula explains the relationship between the probability of the hypothesis before seeing the evidence that p amzn and the probability of the hypothesis after getting the evidence p amzn djia given a hypothesis for amazon given evidence in the dow as a numerical example imagine there is a drug test that is 98 accurate meaning that 98 of the time it shows a true positive result for someone using the drug and 98 of the time it shows a true negative result for nonusers of the drug next assume 0 5 of people use the drug if a person selected at random tests positive for the drug the following calculation can be made to determine the probability the person is actually a user of the drug where the terms are the formula would look like this using the values the calculation works out as follows bayes theorem shows that even if a person tested positive in this scenario there is a 19 76 chance the person takes the drug and an 80 24 chance they don t 