l09_cond_prob

l09_cond_prob - Conditional Probability Reading Ross Ch 3...

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02/23/2009 CS206 - Intro. to Discrete Structures II 1 Conditional Probability Reading: Wednesday, February 25, 2009
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02/23/2009 CS206 - Intro. to Discrete Structures II 2 Example Two dice, 36 equally likely outcomes. E 1 : first dice is 3. E 2 : sum of two dice = 6. P(E 1 )= ? P(E 2 ) = ? P(E 2 if E 1 happened) = ? (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) E 1 E 2 Wednesday, February 25, 2009
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02/23/2009 CS206 - Intro. to Discrete Structures II 3 Example (cont’d) Ω = {(1,1),…,(6,6)} E 1 = {(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)} E 2 = {(5,1),(4,2),(3,3),(2,4),(1,5)} E 1 E 2 = {(3,3)} P(E 1 ) = |E 1 |/| Ω | = 6/36 = 1/6 P(E 2 ) = |E 2 |/| Ω | = 5/36 P(E 1 ,E 2 ) = P(E 1 E 2 ) = |E 1 E 2 |/| Ω |=1/36 Wednesday, February 25, 2009
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02/23/2009 CS206 - Intro. to Discrete Structures II 4 Example (cont’d) P(E 2 if E 1 happened) = ? Ω n = Ω if E 1 happened = E 1 E n = E 2 if E 1 happened = E 2 E 1 = {(3,3)} P(E 2 if E 1 happened) = |E n | / | Ω n | = |E 2 E 1 | / |E 1 | = 1/6 Notation: P(E 2 if E 1 happened) = P(E 2 | E 1 ) – “E 1 given E 2 Given Wednesday, February 25, 2009
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02/23/2009 CS206 - Intro. to Discrete Structures II 5 Example (cont’d) P(E 2 | E 1 ) = |E 2 E 1 | / |E 1 | = ( |E 2 E 1 | / | Ω | ) / ( |E 1 | / | Ω | ) = P(E 1 ,E 2 ) / P(E 1 ) Wednesday, February 25, 2009
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02/23/2009 CS206 - Intro. to Discrete Structures II 6 Conditional Probability Let there be two events, A and B µ Ω . The conditional probability of A|B is P(A|B) = P(A,B) / P(B) when P(B)>0. • A (or B) – consequence B (or A) – cause Wednesday, February 25, 2009
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02/23/2009 CS206 - Intro. to Discrete Structures II 7 Example I Assume you have a bin with 25 light bulbs. Of those, 5 are good (will last over 2 months), 10 are cheap (will last less
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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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l09_cond_prob - Conditional Probability Reading Ross Ch 3...

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