Bayes’ theorem, Mathematics

Bayes’ Theorem

In its general form, Bayes' theorem deals with specific events, such as A1, A2,...., Ak, that have prior probabilities. These events are mutually exclusive events that cover the entire sample space. Each prior probability is already known to the decision maker, and these probabilities have the following form: P(A1), P(A2),...., P(Ak). The events with prior probabilities produce, cause, or precede another event, say B.
A conditional probability relation exists between events A1, A2, ....., Ak and event B. The conditional probabilities are P(B|A1), P(B|A2), ..., P(B|Ak).

Bayes' formula allows us to calculate the probability of an event, say, A1 occurring given that event B has already occurred with a known probability, P(B). The probability of A1 occurring given that B has already occurred is the posterior (or revised) probability. It is denoted by P(A1|B). Thus, we are given P(A1) and the P(B|A1) which we use to calculate P(A1|B).

For any event Ai, Bayes' theorem has the form

2115_bayes theorem.png

 The probability that A1 and B occur simultaneously is equal to the probability that A1 occurs multiplied by the probability that B occurs given A1. Thus, we have

P(A1 and B) = P(A1) P(B|A1)

Since A1, A2, . . . . , Ak form a partition of the entire sample space when event B occurs, only one of the events in the partition occurs. Thus, we have

P(B) = P(A1 and B) + P(A and B) + .... + P(Ak and B)

We already know that for any event Ai,

P(Ai and B) = P(Ai) P(B|Ai)

When we substitute the formula for P(Ai and B) into the equation for P(B) we obtain

P(B) = P(A1) P(B|A1) + P(A2) P(B|A2) +...+ P(Ak) P(B|Ak)

If we then substitute P(B) and P(Ai and B) into the conditional probability, i.e. P(A|B) =  623_bayes theorem1.png    we obtain the generalized version of Bayes' formula, which is shown in the box.

Bayes' Theorem

P(Ai | B)  = 419_bayes theorem2.png

 

Example 

Suppose that a personnel administrator wishes to hire one person from among a number of job applicants for a clerical position. The job to be filled is fairly simple. On the basis of past experience, the personnel director feels that there is a 0.80 probability of an applicant being able to fill the position. This probability is the prior probability.

A personnel administrator usually interviews or tests each applicant, rather than select one at random. This procedure supplies additional direct information about the applicant. In light of this additional information, the personnel director may revise the prior probability about an applicant's chances for success or failure at the job. The revised probability is the posterior probability.

The terms prior and posterior refer to the time when information is collected. Before information is obtained, we have prior probabilities. Bayes' theorem provides a means of calculating posterior probabilities from prior probabilities. The next example illustrates the use of Bayes' theorem.

Posted Date: 9/14/2012 4:49:57 AM | Location : United States







Related Discussions:- Bayes’ theorem, Assignment Help, Ask Question on Bayes’ theorem, Get Answer, Expert's Help, Bayes’ theorem Discussions

Write discussion on Bayes’ theorem
Your posts are moderated
Related Questions
Bob is 2 years from being double as old as Ellen. The sum of twice Bob's age and three times Ellen's age is 66. How old is Ellen? Let x = Ellen's age and let y = Bob's age. Sin

Consider the integral where the notation means a contour that is parallel to the real z axis, but moved down by a distance d . Use the method of steepest descents to deri

what are challenges and solution of international marketing

It's easier to describe an explicit solution, in this case and then tell you what an implicit solution is not, and after that provide you an illustration to demonstrate you the dif

in regrouping if we have abig number in the end what should i do?add an number on top of it,please help


If a mean score is 89 with a standard deviation of 8 points. What is the least score you can make and be in the top 20%?

The product of -7ab and +3ab is (-7 x 3) a 2  b 2  = -21a 2  b 2 . In other words, a term with minus sign when multiplied with a term having a positive sign, gives a product having

Determine the Probability From a pack of playing cards what is the probability of; (i)  Picking either a 'Diamond' or a 'Heart' → mutually exclusive (ii) Picking either

Maya says thafl for instance, to help the children of Class 2 construct the '5 times table', she uses their hands. Each child counts how many fingers on one hand, and then how ma