Bayes Theorem

Bayes’ TheoremBy SabareeshBabu and Rishabh Kumar

IntroductionShows the relation between one conditional probability and its inverse.Provides a mathematical rule for revising an estimate or forecast in light of experience and observation. Relates-Prior Probability of A, P(A), is the probability of event A not concerning its associated event B - Prior Probability of B, P(B), is the probability of B not concerning A - Conditional Probability of B given A, P(B│A). Also called the likelihood-Conditional Probability of A given B, P(A│B). Also called the posterior probability.

Simple example of prior, conditional, and posterior probabilityThe prior probability of A, P(A), is 1/6. The prior probability of B, P(B), is 1/6. The prior probability of C, P(C), is 5/36

The conditional probability of event C given that A occurs, P(C│A), is 1/6

The posterior probability, P(A│C), is 1/5 What is P(C│B)? Answer: 0

Example 11% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

Without Bayes’ TheoremCreate a large sample size and use probabilities given in the problem to work out the problem. Assume, for example, that 10,000 women participate in a routine screening for breast cancer. 1%, or 100 women, have breast cancer. 80% of women with breast cancer, 80 women, will get positive mammographies. 9.6%,950 women, of the 9900 women who don’t have breast cancer will also get positive mammographies. Create a table using the numbers obtained from the assumed sample size and determine the answer.

Without Bayes’ Theorem cont.1030 women950 women80 women8970 women8950 women20 women10000 women9900 women100 womenOut of the 1030 women who get positive mammographies only 80 actually have breast cancer, therefore, the probability is 80/1030 or 7.767%

Bayes’ Theorem:The posterior probability is equal to the conditional probability of event B given A multiplied by the prior probability of A, all divided by the prior probability of B.

Using Bayes’ Theorem1% of women at age forty who participate in routine screening have breast cancer. P(B)= 0.01 80% of women with breast cancer will get positive mammographies. P(+│B) = 0.8 9.6% of women without breast cancer will also get positive mammographies. P(+│B’) = 0.096 A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer? Find P(B│+)

Using Bayes’ Theorem cont. P(B│+) = P(+│B) P(B) P(+) P(B), P(+│B), and P(+│B’) are known. P(+) is needed to find P(B│+) P(+) = P(+│B) P(B) + P(+│B’) P(B’) P(+) = (0.8) ( 0.01) + (0.096) (0.99) P(+) = 0.1030 P(B│+) = (0.8) (0.01) (0.1030) P(B│+) = 0.07767

Example 2Two different suppliers, A and B, provide a manufacturer with the same part. All supplies of this part are kept in a large bin. in the past, 5% of the parts supplied by A and 9% of the parts supplied by B have been defective. A supplies four times as many parts as B Suppose you reach into the bin and select a part, and find it is nondefective. What is the probability that it was supplied by A?

Solution5% of the parts supplied by A and 9% of the parts supplied by B have been defective. P (D│A) = 0.95 P(D│B) = 0.91 A supplies four times as many parts as B P(A) = 0.8 P(B) = 0.2 Suppose you reach into the bin and select a part, and find it is nondefective. What is the probability that it was supplied by A?Find P(A│D)

Solution cont. P(A│D) = P(D│A) P(A) P(D) P (D│A) = 0.95 P(A) = 0.8 P(D)= P (D│A) P(A) + P(D│B) P(B) P(D) = (0.95) (0.8) + (0.91) (0.2) = 0.942 P(A│D) = (0.95) (0.8) (0.942) P(A│D) = 0.8068

Bayes Theorem

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