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Unformatted text preview: 1 Dice – Our Misunderstood Friends • Roll two 6sided dice, yielding values D 1 and D 2 • Let E be event: D 1 + D 2 = 4 • What is P(E)? S = 36, E = {(1, 3), (2, 2), (3, 1)} P(E) = 3/36 = 1/12 • Let F be event: D 1 = 2 • P(E, given F already observed)? S = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)} E = {(2, 2)} P(E, given F already observed) = 1/6 Conditional Probability • Conditional probability is probability that E occurs given that F has already occurred “Conditioning on F” • Written as P(E  F) Means “P(E, given F already observed)” Sample space, S, reduced to those elements consistent with F (i.e. S F) Event space, E, reduced to those elements consistent with F (i.e. E F) • With equally likely outcomes: P(E  F) = = # of outcomes in E consistent with F # of outcomes in S consistent with F         F EF SF EF Conditional Probability • General definition: where P(F) > 0 • Holds even when outcomes are not equally likely • Implies: P(EF) = P(E  F) P(F) (chain rule) • What if P(F) = 0? P(E  F) undefined Congratulations! You observed the impossible! ) ( ) ( )  ( F P EF P F E P Generalized Chain Rule • General definition of Chain Rule: • Ross calls this the “multiplication rule” • You can call it either (just be consistent) ) ... ( 3 2 1 n E E E E P ) ...  ( )...  ( )  ( ) ( 1 2 1 2 1 3 1 2 1 n n E E E E P E E E P E E P E P Slicing Up the Spam • 24 emails are sent 6 each to 4 users. • 10 of the 24 emails are spam. All possible outcomes equally likely E = user 1 receives 3 spam emails What is P(E)? 3 2 4 5 . 6 2 4 3 1 4 3 1 0 Slicing Up the Spam • 24 emails are sent 6 each to 4 users. • 10 of the 24 emails are spam. All possible outcomes equally likely E = user 1 receives 3 spam emails F = user 2 receives 6 spam emails What is P(E  F)?...
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This document was uploaded on 12/24/2011.
 Spring '09

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