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# Bayes’ Theorem – Definition

### Bayes’ Theorem’ Definition

Bayes’ theorem refers to a mathematical formula used to determine conditional probability. The theorem was named after Thomas Bayes, an 18th-century British mathematician. This theorem offers a method of revising existing theories or predictions given new or even additional evidence. Bayes’ theorem, in finance, can be utilized in rating the risk involved in lending money to potential borrowers.

The formula goes thus:

Bayes’ theorem is also referred to as Bayes’ Law or Bayes’ Rule.

### A Little More on What is the Bayes’ Theorem

The theorem’s applications are extensive and not restricted to the financial sphere. For instance, Bayes’ theorem can be utilized in determining how accurate medical test results are by considering how possible any specific individual is to have a disease, as well as, the test’s general accuracy.

Bayes’ theorem gives the likelihood of an event dependent on the information which is or might be related to that event. The formula can be utilized in seeing how the probability of an event happening is affected by entirely new information if the new information is true. For instance, say one card is drawn from a full deck of 52 cards. The probability of the card being a king is 4 divided by 52, which is equal to 1/13 or approximately 7.69%. Keep in mind that 4 kings exist in the deck card. Assume it’s revealed that the chosen card is a face card. The probability of the selected card being a king, given it’s a face card, is 4 divided by 12, or approximately 33.3%, as a deck has 12 face cards.

Bayes’ theorem follows from the principle of conditional probability. Conditional probability refers to the probability of an event considering that another event occurred. For instance, an easy probability question might be “What’s the probability of Amazon.com, Inc., (AMZN) stock price falling? ” This question is taken a step further by conditional probability, in that it asks “What’s the probability of Amazon stock price falling considering the fact that the Dow Jones Industrial Average index had fallen earlier?”

A’s conditional probability given considering that B has occurred can be expressed thus:

P(A|B) = P(A and B) / P(B) = P(A∩B) / P(B)

If A stands for Amazon price falls and B, DJIA is already down, the conditional probability expression would read as “the probability that Amazon drops given a decline in DJIA equals the probability that Amazon price declines and also DJIA falls over the probability of a DJIA index decrease.

The probability of A, as well as, B occurring is P(A∩B). It’s the same as the probability of A occurring multiplied by the probability that B occurs considering that A occurred, shown as P(A) x P(B|A). Making use of the same rationale, P(A∩B) is also the probability that B occurs multiplied by the probability that A occurs considering that B occurs, shown as P(B) x P(A|B). The fact that the two expressions are equal brings about the Bayes’ theorem and it’s written as:

if P(A∩B) = P(A) x P(B|A) = P(B) x P(A|B)

then, P(A|B) = [P(A) x P(B|A)] / P(B).

Where P(A) and P(B) are A and B’s probabilities with no regard to each other.

P(B|A) is the probability of B occurring given A is true.

Finally, the conditional probability of A occurring given that B is true is P(A|B).

This formula explicates the relationship existing between the hypothesis’ probability before getting the evidence P(A) and then the hypothesis’ probability after getting the evidence P(A|B), given hypothesis A, as well as, evidence B.

Another instance, imagine that a drug test exists which is 98% accurate which means that 98% of the time its result is positive for someone taking the drug and its result is negative 98% of the time for nonusers of the drug. Next, assume the drug is used by 0.5% of the people. If someone selected randomly, tests positive to the drug, the calculation below can be made to ascertain the probability of the person being an actual user of that drug.

(0.98 x 0.005) / [(0.98 x 0.005) + ((1 – 0.98) x (1 – 0.005))] = 0.0049 / (0.0049 + 0.0199) = 19.76%.

References for Bayesian Theorem