l10_bayes_rule

l10_bayes_rule - Bayes' Rule Reading: Ross, Ch 3., Sec. 3....

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02/25/2009 CS206 - Intro. to Discrete Structures II 1 Bayes’ Rule Reading: Ross, Ch 3., Sec. 3. Wednesday, February 25, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 2 Complementary causes Let E and C be two events µ Ω . Then: P(E) = P(E|C)P(C) + P(E|C c )P(C c ) Ω E C C c Wednesday, February 25, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 3 Example: two explanations Assume there are two dice. One is fair, the other one is biased and always lands on 3. You pick a die and roll it. What is the probability of getting 3? Both dice are equally likely to be picked. A = roll die 1, A c = roll die 2 P(A) = P(A c ) = 0.5 P(3|A) = 1/6, P(3|A c ) = 1 P(3) = ? P(3) = P(3,A [ A c ) = P(3,A) + P(3,A c ) = P(3|A)P(A) + P(3|A c )P(A c ) = 1/6 ½ + 1 ½ = 7/12 3 is a consequence of rolling either A or A c A and A c are complementary Wednesday, February 25, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 4 Complementary causes - proof • Proof: P(E) = P(E Ω ) = P(E (C C c )) = P( (E C) (E C c ) ) = P(E,C) + P(E,C c ) = P(E|C)P(C) + P(E|C c )P(C c ) (since E C and E C c are mutually exclusive) Wednesday, February 25, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 5 Example 1 In answering a question on a multiple-choice test a student either knows the answer or guesses. Let p=0.8 be the probability that the student knows the answer and 1-p=0.2 the probability that the student guesses. Assume that if the student guesses the answer, she will be correct with probability 1/m, where m is the number of choices/ alternative answers. Suppose m=4. What is the probability that the student will answer a question correctly? C = student answers question correctly K = student knows the answer, P(K) = p = 0.8. K c = student does not know the answer. P(K c ) = 1-p=0.2. P(C) = ? Wednesday, February 25, 2009
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CS206 - Intro. to Discrete Structures II 6 Example 1 (cont’d) • P(C) = P(C|K)P(K) + P(C|K c )P(K c ) P(C|K) = 1 P(C|K c ) = 1/m = 0.25. P(C) = 1
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This note was uploaded on 03/24/2011 for the course CS 206 taught by Professor Fredman during the Spring '08 term at Rutgers.

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l10_bayes_rule - Bayes' Rule Reading: Ross, Ch 3., Sec. 3....

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