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TOTAL PROBABILITY AND BAYES’ THEOREM EXAMPLE 1. 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). Bayer's Theorem Examples with Solutions. Bayes theorem gives a relation between P(A|B) and P(B|A). 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. Example: Bayes' theorem to find conditional porbabilities is explained and used to solve examples including detailed explanations. Also the numerical results obtained are discussed in order to understand the possible applications of the theorem. 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. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the ﬁrst head is observed. For any event B Pr(AijB) = Pr(Ai)Pr(BjAi) Σn j=1 Pr(Aj)Pr(BjAj): † Proof. Diagrams are used to give a visual explanation to the theorem. We will look at four di erent versions of Bayes rule for random vari-ables. 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. We use Bayes’ theorem every time we calculate these values from a 2x2 table, even though it does not feel like it. Let A1;:::;An be a partition of Ω. Compute the probability that the ﬁrst head appears at an even numbered toss. Bayes’ Theorem † Bayes Theorem. Pr(AijB) = Pr(Ai \B) Pr(B) = Pr(Ai)Pr(BjAi) Σn j=1 Pr(Aj)Pr(BjAj): † Example. As a formal theorem, Bayes’ theorem is valid in all interpretations of prob-ability. 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.