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 occurring. 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.
Bayes' theorem is also called Bayes' Rule or Bayes' Law and is the foundation of the field of Bayesian statistics.

Applications of the theorem are widespread and not limited to the financial realm. As an 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. Prior probability, in Bayesian statistical inference, is the probability of an event before new data is collected. This is the best rational assessment of the probability of an outcome based on the current knowledge before an experiment is performed. Posterior probability is the revised probability of an event occurring after taking into consideration new information. Posterior probability is calculated by updating the prior probability by using Bayes' theorem. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred.

Bayes' theorem thus gives the probability of an event based on new information that is, or may be related, to that event. The formula can also be used to see how the probability of an event occurring is affected by hypothetical new information, supposing the new information will turn out to be true. For instance, say a single card is drawn from a complete deck of 52 cards. The probability that the card is a king is four divided by 52, which equals \(\displaystyle\frac{1}{{13}}\) or approximately \(7.69\%\). Remember that there are four kings in the deck. 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' Theorem

\(\displaystyle{P}{\left({A}{|}{B}\right)}=\frac{{{P}{\left({A}\bigcap{B}\right)}}}{{{P}{\left({B}\right)}}}={\left({P}{\left({A}\right)}\cdot{P}{\left({B}{|}{A}\right)}\right)}{\left({P}{\left({B}\right)}\right)}\)

where

\(\displaystyle{P}{\left({A}\right)}=\) The probability of A occuring

\(\displaystyle{P}{\left({B}\right)}=\) The probability of B occuring

\(\displaystyle{P}{\left({A}{|}{B}\right)}=\) The probability of A given B

\(\displaystyle{P}{\left({B}{|}{A}\right)}=\) The probability of B given A

\(\displaystyle{P}{\left({A}\bigcap{B}\right)}=\) The probability of both A and B occuring

Applications of the theorem are widespread and not limited to the financial realm. As an 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. Prior probability, in Bayesian statistical inference, is the probability of an event before new data is collected. This is the best rational assessment of the probability of an outcome based on the current knowledge before an experiment is performed. Posterior probability is the revised probability of an event occurring after taking into consideration new information. Posterior probability is calculated by updating the prior probability by using Bayes' theorem. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred.

Bayes' theorem thus gives the probability of an event based on new information that is, or may be related, to that event. The formula can also be used to see how the probability of an event occurring is affected by hypothetical new information, supposing the new information will turn out to be true. For instance, say a single card is drawn from a complete deck of 52 cards. The probability that the card is a king is four divided by 52, which equals \(\displaystyle\frac{1}{{13}}\) or approximately \(7.69\%\). Remember that there are four kings in the deck. 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' Theorem

\(\displaystyle{P}{\left({A}{|}{B}\right)}=\frac{{{P}{\left({A}\bigcap{B}\right)}}}{{{P}{\left({B}\right)}}}={\left({P}{\left({A}\right)}\cdot{P}{\left({B}{|}{A}\right)}\right)}{\left({P}{\left({B}\right)}\right)}\)

where

\(\displaystyle{P}{\left({A}\right)}=\) The probability of A occuring

\(\displaystyle{P}{\left({B}\right)}=\) The probability of B occuring

\(\displaystyle{P}{\left({A}{|}{B}\right)}=\) The probability of A given B

\(\displaystyle{P}{\left({B}{|}{A}\right)}=\) The probability of B given A

\(\displaystyle{P}{\left({A}\bigcap{B}\right)}=\) The probability of both A and B occuring