l11_indep

L11_indep - Independence Reading Ross Ch 3 Sec 4 Monday March 9 2009 CS206 Intro to Discrete Structures II 1 Example 1 From a deck of 52 cards we

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02/25/2009 CS206 - Intro. to Discrete Structures II 1 Independence Reading: Ross, Ch 3., Sec. 4. Monday, March 9, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 2 Example 1 From a deck of 52 cards we draw one card randomly. What is the probability that this card will be an ace? What is the probability that it will be a spade? Finally, what is the probability that it will be an ace spade? A – card is an ace S – card is a spade A,S – card is an ace spade P(A) = 4/52, P(S) = 13/52, P(A,S) = 1/52 P(A) ¢ P(S) = 4/52 ¢ 13/52 = 1/52 P(A)P(S) = P(A,S). Interesting? What is so specific about A & S? They are independent! Monday, March 9, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 3 Example 2 • Recall Investigation example from Bayes’ Rule lecture. When the proportion of brown-haired people was 100%, the evidence did not matter: P(G|E) = 0.6 = P(G) ) “G is independent of E” When the proportion was 20%, the evidence did matter: P(G|E) = 0.88 P(G) ) “G depends on E” Monday, March 9, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 4 Independence Let A and B be two events µ Ω . If P(A,B) = P(A) ¢ P(B) A & B are said to be independent . Otherwise, they are dependent . P(A|B) = P(A) or P(B|A) = P(B). Prove it! Monday, March 9, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 5 Example 3 • In Example 1 we showed that A & S are independent. What can one say about A and S c ? A S c – a card is an ace and it is not a spade P(A,S c ) = 3/52 P(A) = 4/52, P(S c ) = 39/52 P(A) P(S c ) = 4/52 39/52 = 3/52 Hence, A and S c are independent as well! Monday, March 9, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II 6 Proposition • If A and B Ω are independent, then (A B c ), (A c B) and (A c B c ), and are also independent. Proof: P(A) = P(A|B)P(B) + P(A,B c ) = P(A)P(B) + P(A,B c ) ) P(A,B c ) = P(A)[ 1 – P(B) ] = P(A)P(B c ) ) c are independent. Proofs for other cases follow directly. Monday, March 9, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II Graphical View of Independence 7 1 2 3 4 5 6 7 8 9 J Q K A Monday, March 9, 2009
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02/25/2009 CS206 - Intro. to Discrete Structures II Graphical View of Independence 7 1 2 3 4 5 6 7 8 9 J Q K A P(spade) = 13/52 Monday, March 9, 2009
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This note was uploaded on 03/24/2011 for the course CS 206 taught by Professor Fredman during the Spring '08 term at Rutgers.

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L11_indep - Independence Reading Ross Ch 3 Sec 4 Monday March 9 2009 CS206 Intro to Discrete Structures II 1 Example 1 From a deck of 52 cards we

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