TOTAL PROBABILITY AND BAYES’ THEOREM EXAMPLE 1. Let A1;:::;An be a partition of Ω. Pr(AijB) = Pr(Ai \B) Pr(B) = Pr(Ai)Pr(BjAi) Σn j=1 Pr(Aj)Pr(BjAj): † Example. An important application of Bayes’ theorem is that it gives a rule how to update or revise the strengths of evidence-based beliefs in light of new evidence a posteriori. In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Reverend Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. Bayes' theorem to find conditional porbabilities is explained and used to solve examples including detailed explanations. Bayes’ Theorem Authors: Blume, Greevy Bios 311 Lecture Notes Page 14 of 26 Diagnostic tests Sensitivity, Specificity, Positive and Negative Predictive Value are all related via Bayes’ Theorem. Compute the probability that the first head appears at an even numbered toss. Example: Proof of Bayes Theorem The probability of two events A and B happening, P(A∩B), is the probability of A, P(A), times the probability of B given that A has occurred, P(B|A). Also the numerical results obtained are discussed in order to understand the possible applications of the theorem. We use Bayes’ theorem every time we calculate these values from a 2x2 table, even though it does not feel like it. Diagrams are used to give a visual explanation to the theorem. As a formal theorem, Bayes’ theorem is valid in all interpretations of prob-ability. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the first head is observed. We will look at four di erent versions of Bayes rule for random vari-ables. Bayes rule for random variables There are many situations where we want to know X, but can only measure a related random variable Y or observe a related event A. Bayes gives us a systematic way to update the pdf for Xgiven this observation. Bayes’ Theorem † Bayes Theorem. For any event B Pr(AijB) = Pr(Ai)Pr(BjAi) Σn j=1 Pr(Aj)Pr(BjAj): † Proof. Bayer's Theorem Examples with Solutions. Bayes theorem gives a relation between P(A|B) and P(B|A).

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