Discussion3

Discussion3 - Discussion 3 Emily Mower February 1, 2010...

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Discussion 3 Emily Mower February 1, 2010 Topics to be covered: Helpful probabilistic concepts Bayes Rule Partitioning the sample space 1 Helpful probabilistic concepts 1.1 Conditional probability The conditional probability of B, given A, denoted by P ( B | A ) is defined by: P ( B | A ) = P ( A B ) P ( A ) , provided P ( A ) > 0 1.2 Multiplicative rules If in an experiment the events A and B can both occur, then: P ( A B ) = P ( A ) P ( B | A ) , provided P ( A ) > 0 This comes from multiplying both sides of the conditional probabilty defini- tion by P ( A ). 1
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2 Partitioning the sample space If the events A 1 , A 2 , ..., A k constitute a partition of the sample space S such that P ( A i ) 6 = 0 for i = 1 , 2 ,...,k , then for any event B of S , P ( B ) = k X i =1 P ( A i B ) = k X i =1 P ( A i ) P ( B | A i ) Note: The events A n ( n = { 1 , 2 ,...,k } ) are disjoint (see Figure 1)! Figure 1: Sample space partition (image source: http://www.math.uah.edu/stat/prob/Total.png) . 2
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Bayes Rule If the events A 1 ,A 2 ,...A k constitute a partition of the sample space S such that P ( A i ) 6 = 0 for i = 1 , 2 ,...,k , then for any event B in S such that P ( B ) 6 = 0, P ( A r | B ) = P ( A r B ) k i =1 P ( A i B ) = P ( A r ) P ( B | A r ) k i =1 P ( A i ) P ( B | A i ) for r = 1 , 2 ,...,k Let’s look at the example for two events, A and B : P ( A | B ) = P ( A B ) P ( B ) P ( B | A ) = P ( A B ) P ( A ) This implies that: P ( A | B ) = P ( B | A ) P ( A ) P ( B ) The formulation above arrives at the same conclusion. Since P ( B ) can also
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Discussion3 - Discussion 3 Emily Mower February 1, 2010...

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