post1 - Probability Properties: Experiment: any action or...

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Probability Experiment: any action or process that gener- ates observations. Sample Space C : set of all possible outcomes of the experiment. (or random phenomenon) Event: any collection of outcomes contained in the sample space C . Axioms 1. P ( A ) 0, for any event A . 2. P ( C )=1. 3. If A 1 ,A 2 , ··· k is a finite collection of mu- tually exclusive events, then P ( A 1 A 2 ∪···∪ A k )= k X i =1 P ( A i ) . (same thing works for infinite collection) 1 Properties: P ( A * )=1 - P ( A ), for any event A . If A and B are mutually exclusive, then P ( A B )=0 . P ( A B P ( A )+ P ( B ) - P ( A B ) . P ( A B C P ( A P ( B P ( C ) - P ( A B ) - P ( A C ) - P ( B C P ( A B C ) . Conditional Probability P ( A | B P ( A B ) P ( B ) ( P ( B ) > 0) Independence P ( A | B P ( A )o r P ( B | A P ( B ) 2 Example (p21) A hand of 5 cards is to be dealt at random without replacement from an ordi- nary deck of 52 playing cards. C 1 = at least 4 spades in the hand C 2 = all-spade hand P ( C 2 | C 1 )=? Example : Two tickets are drawn at random from the box containing 1 , 2 , 3 ,and 4 . C 1 = 2nd ticket is 4 C 2 = 1st ticket is 2 w/o replacement, w/ replacement 3 Example : John and Michael go duck hunting together. Sps that John hits the target w/p 0.3 and Michael, independently, w/p 0.1. They both fire one shot at a duck. a) Given that exactly one shot hits the duck, what is the conditional prob. that it is John’s shot? Michael’s? b) Given that the duck is hit, what is the con- ditional prob. that John hit it? Michael hit it? 4
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Bayes’ Theorem P ( A j | B )= P ( A j B ) P ( B ) = P ( B | A j ) P ( A j ) k i =1 P ( B | A i ) P ( A i ) for j =1 , ··· ,k .( A i ’s are a partition of C ) Example : A blood test is 99% effective in detecting a certain disease when the disease is present. However, the test also yields a false-positive result for 2% of healthy patients tested. Suppose 0.1% of the population has the disease. What is the probability that a ran- domly tested individual actually has the disease given that his or her test result is positive?
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post1 - Probability Properties: Experiment: any action or...

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