ch3-review - Chapter 3. • Conditional probability...

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Unformatted text preview: Chapter 3. • Conditional probability definition: P (A ∩ B ) . P (A|B ) = P (B ) • Multiplication rule P (A ∩ B ) = P (B ) · P (A|B ) = P (A) · P (B |A). • Definition of a partition of S . • If A1 , A2 , . . . , An is a partition of S (a special case being A and Ac ) then rule of average conditional probabilities P (B ) = P (B |A1 )P (A1 ) + . . . + P (B |An )P (An ). • Bayes formula If A1 , A2 , . . . , An is a partition of S and we know P (B |Ai ) for each i, then we use Bayes formula to find P (Ai |B ): P (B |Ai )P (Ai ) P (Ai |B ) = ∀i. P (B |A1 )P (A1 ) + . . . + P (B |An )P (An ) • Independence. If A and B are independent events, then P (A ∩ B ) = P (A) · P (B ). Comparing with multiplication rule, observe that P (A|B ) = P (A) and P (B |A) = P (B ) if A and B are independent. Extend this idea of independence to more than two events. If A1 , A2 , . . . An are independent then P (A1 ∩ · · · An ) = P (A1 ) · P (A2 ) · · · P (An ), however this is not a sufficient condition for independence of n events. 1 ...
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