Math 203_Lecture 4

Math 203_Lecture 4 - Principles of Statistics 1 Lecture 4:...

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Unformatted text preview: Principles of Statistics 1 Lecture 4: Math 203 Abbas Khalili Department of Mathematics and Statistics McGill University May 05, 2011 Principles of Statistics 1 Lecture 4: Math 203 p. 1/ ? ? Bayes Theorem Sometimes we know the conditional probability P ( A | B ) and would like to know about the conditional probability P ( B | A ) , or vice versa. Principles of Statistics 1 Lecture 4: Math 203 p. 2/ ? ? Bayes Theorem Sometimes we know the conditional probability P ( A | B ) and would like to know about the conditional probability P ( B | A ) , or vice versa. Bayes theorem helps us in these situations. Principles of Statistics 1 Lecture 4: Math 203 p. 2/ ? ? Bayes Theorem Sometimes we know the conditional probability P ( A | B ) and would like to know about the conditional probability P ( B | A ) , or vice versa. Bayes theorem helps us in these situations. Consider the medical example discussed before. Principles of Statistics 1 Lecture 4: Math 203 p. 2/ ? ? Bayes Theorem Sometimes we know the conditional probability P ( A | B ) and would like to know about the conditional probability P ( B | A ) , or vice versa. Bayes theorem helps us in these situations. Consider the medical example discussed before. Given that the test result for a person is +, what is the probability that the person is actually a carrier? Principles of Statistics 1 Lecture 4: Math 203 p. 2/ ? ? Bayes Theorem Sometimes we know the conditional probability P ( A | B ) and would like to know about the conditional probability P ( B | A ) , or vice versa. Bayes theorem helps us in these situations. Consider the medical example discussed before. Given that the test result for a person is +, what is the probability that the person is actually a carrier? We want: P ( C | +) Principles of Statistics 1 Lecture 4: Math 203 p. 2/ ? ? Bayes Theorem Sometimes we know the conditional probability P ( A | B ) and would like to know about the conditional probability P ( B | A ) , or vice versa. Bayes theorem helps us in these situations. Consider the medical example discussed before. Given that the test result for a person is +, what is the probability that the person is actually a carrier? We want: P ( C | +) Note that in the example we are given P (+ | C ) = . 99 . Principles of Statistics 1 Lecture 4: Math 203 p. 2/ ? ? Bayes theorem in this example says: P ( C | +) = P (+ | C ) P ( C ) P (+ | C ) P ( C ) + P (+ | C c ) P ( C c ) = . 99 . 07 . 99 . 07 + . 025 . 93 = . 0693 . 09255 . 749 Principles of Statistics 1 Lecture 4: Math 203 p. 3/ ? ? Bayes theorem in this example says: P ( C | +) = P (+ | C ) P ( C ) P (+ | C ) P ( C ) + P (+ | C c ) P ( C c ) = . 99 . 07 . 99 . 07 + . 025 . 93 = . 0693 . 09255 . 749 Notice how different the two probabilities P (+ | C ) and P ( C | +) are in this example....
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Math 203_Lecture 4 - Principles of Statistics 1 Lecture 4:...

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