Bayes' Theorem

What is the 'Bayes' Theorem'

Bayes' theorem is a mathematical formula for determining conditional probability named after 18th-century British mathematician Thomas Bayes. The theorem provides a way to revise existing predictions or theories given new or additional evidence. In finance, Bayes' theorem can be used to rate the risk of lending money to potential borrowers.

The formula is as follows:

Also called "Bayes' Rule."

BREAKING DOWN 'Bayes' Theorem'

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 gives the probability of an event based on information that is or may be related to that event. The formula can be used to see how the probability of an event occurring is affected by new information, supposing the new information is true. For example, say a single card is drawn from a complete deck of 52 cards. The probability the card is a king is four divided by 52, or approximately 7.69%, since there are four kings in the deck. Now, suppose it is revealed 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%, since there are 12 face cards in a deck.

Bayes' Theorem Formula

The formula is written as P(A|B) = P(B|A) * P(A) / P(B). P(A) and P(B) are the probabilities of A and B without regard to each other. P(B|A) is the probability that B will occur given A is true. Finally, the answer, P(A|B) is the conditional probability of A occurring given B is true.

As another example, imagine there is a drug test that is 98% accurate, meaning 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 see the probability the person is actually a user of the drug.

(0.98 * 0.005) / ((0.98 * 0.005) + (0.02 * 0.995)) = 0.0049 / (0.0049 + 0.0199) = 19.76%

Bayes' theorem shows that even if a person tested positive in this scenario, it is actually much more likely the person is not a user of the drug.