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.

RELATED TERMS
  1. Posterior Probability

    The revised probability of an event occurring after taking into ...
  2. Coase Theorem

    A legal and economic theory that affirms that where there are ...
  3. Irrelevance Proposition Theorem

    A theory of corporate capital structure that posits that financial ...
  4. Mutual Fund Theorem

    An investing theory, postulated by Nobel laureate James Tobin, ...
  5. Arrow's Impossibility Theorem

    A social-choice paradox illustrating the impossibility of having ...
  6. Fisher's Separation Theorem

    A theory stating that: 1. A firm's choice of investments are ...
Related Articles
  1. Trading

    Financial Forecasting: The Bayesian Method

    This method can help refine probability estimates using an intuitive process.
  2. Investing

    Understanding the Modigliani-Miller Theorem

    The Modigliani-Miller (M&M) theorem is used in financial and economic studies to analyze the value of a firm, such as a business or a corporation.
  3. Markets

    Explaining the Central Limit Theorem

    Central limit theorem is a fundamental concept in probability theory.
  4. Markets

    Explaining the Coase Theorem

    The Coase theorem states when there are competitive markets and no transaction costs, bargaining will lead to a mutually beneficial outcome.
  5. Markets

    Why the Bay Area's Economy is on Fire (for Now)

    The Bay Area's economy has the strongest job market and wage growth in the country. How long will it last?
  6. Investing

    How Pharmaceutical Companies Price Their Drugs

    Learn more about how pharmaceutical companies price drugs, why prices are often very high and why it can be difficult to settle on a suitable price.
  7. Personal Finance

    Credit vs. Debit Cards: Which Is Better?

    Be strategic about the card you choose
  8. Personal Finance

    Credit, Debit And Charge: Sizing Up The Cards In Your Wallet

    Not all plastic is equal! Learn the difference between the three kinds, and how each can affect your finances.
  9. Personal Finance

    Why Do Credit Cards Expire?

    Credit cards expire for more reasons than you could imagine – including, so you don't forget you have the card.
  10. Personal Finance

    How Many Credit Cards Should You Have?

    Having several credit cards provides many benefits. Here are a few factors to consider when you’re deciding how many credit cards you need.
RELATED FAQS
  1. What are the main principles of the Heckscher-Ohlin Model?

    Learn about the four main principles of the Heckscher-Olin model, and find out how the model describes patterns of commerce ... Read Answer >>
  2. What is Fisher's separation theorem?

    Fisher's separation theorem stipulates that the goal of any firm is to increase its value to the fullest extent, regardless ... Read Answer >>
  3. I have 3 credit card debts, how can I get them paid off the cheapest way?

  4. What are the primary risks associated with investing in the drugs sector?

    Learn more about the primary risks facing investors in the pharmaceutical and biotechnology industries. Find out how regulation ... Read Answer >>
  5. What process does a company need to follow to bring a new drug to market?

    Learn about the costly price of bringing new drugs to market. Discover why the pharmaceutical industry invests billions of ... Read Answer >>
  6. Do financial advisors get drug tested?

    Learn how financial advisor regulatory bodies do not require drug testing but many individual firms that hire advisors do ... Read Answer >>
Hot Definitions
  1. AAA

    The highest possible rating assigned to the bonds of an issuer by credit rating agencies. An issuer that is rated AAA has ...
  2. GBP

    The abbreviation for the British pound sterling, the official currency of the United Kingdom, the British Overseas Territories ...
  3. Diversification

    A risk management technique that mixes a wide variety of investments within a portfolio. The rationale behind this technique ...
  4. European Union - EU

    A group of European countries that participates in the world economy as one economic unit and operates under one official ...
  5. Sell-Off

    The rapid selling of securities, such as stocks, bonds and commodities. The increase in supply leads to a decline in the ...
  6. Brazil, Russia, India And China - BRIC

    An acronym for the economies of Brazil, Russia, India and China combined. It has been speculated that by 2050 these four ...
Trading Center