Conditional

# Conditional - I NTRODUCTION TO P ROBABILITY T HEORY...

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I NTRODUCTION TO P ROBABILITY T HEORY Conditional Probability Conditional Probability: of event C 2 given the event C 1 , provided that P ( C 1 ) > 0 ,, P ( C 2 | C 1 ) = P ( C 1 C 2 ) P ( C 1 ) Properties: P ( C 2 | C 1 ) 0 P ( C 2 C 3 ∪ · · · | C 1 ) = P ( C 2 | C 1 ) + P ( C 3 | C 1 ) + · · · , provided that C 2 , C 3 , . . . are mutually exclusive. P ( C 1 | C 1 ) = 1

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I NTRODUCTION TO P ROBABILITY T HEORY Examples 5 cards from 52 cards: conditional probability of an all spade hand ( C 2 ) given that there are at least 4 spades in hand ( C 1 ), P ( C 2 | C 1 ) = P ( C 1 C 2 ) P ( C 1 ) = P ( C 2 ) P ( C 1 ) = ( 13 5 ) / ( 52 5 ) [ ( 13 4 )( 39 1 ) + ( 13 5 ) ] / ( 52 5 ) = 0 . 0441 .
I NTRODUCTION TO P ROBABILITY T HEORY Multiplication Rule Multiplication Rule: P ( C 1 C 2 ) = P ( C 1 ) P ( C 2 | C 1 ) 52 cards: Compute the probability that the third spade appears on the sixth draw. C 1 —the event of two spades in the first five draws; C 2 —the event of a spade on the sixth draw; we want P ( C 1 C 2 ) : P ( C 1 C 2 ) = P ( C 1 ) P ( C 2 | C 1 ) = ( 13 2 )( 39 3 ) ( 52 5 ) × 11 47 = 0 . 0642 P ( C 1 C 2 C 3 ) = P (( C 1 C 2 ) C 3 ) = P ( C 1 C 2 ) P ( C 3 | C 1 C 2 ) = P ( C 1 ) P ( C 2 | C 1 ) P ( C 3 | C 1 C 2 )

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I NTRODUCTION TO P ROBABILITY T HEORY Bayes’ Theorem Law of Total Probability: Suppose k mutually exclusive and exhaustive events C 1 , C 2 , . . . , C k such that P ( C i ) > 0 and C = C 1 C 2 ∪ · · · ∪ C k , we have P ( C ) = k X i =1 P ( C i ) P ( C |
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