COMP 170 L2 L16: Conditional Probability and Independence Page 1

COMP 170 L2 Outline Page 2 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

COMP 170 L2 Example: Roll Two Dice

COMP 170 L2 Event F and Change of Sample Spaces

COMP 170 L2 l F favors none of the three remaining possible outcomes l New probabilities n Preserve ratios n Sum to one

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Outline Page 10 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

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A Note

COMP 170 L2 Use of Conditional Probability Page 15

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Outline Page 17 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

COMP 170 L2 Motivating Example Page 18 l Learning that F occurred does not change probability of E l F is irrelevant to E l E is independent of F

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COMP 170 L2 Outline Page 21 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

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COMP 170 L2 Outline Page 24 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

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COMP 170 L2 Outline Page 29 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

COMP 170 L2 Multiple Stage Process l Recall l Probability associated with a process with uncertain outcomes l Sample space: set of all possible outcomes l Event: subset of sample space l Probability distribution: weight for each element l One stage process l Flipping a coin once l Throwing two dice (all at once) l Multiple stage process l Flip coin multiple times l Hashing: Hash keys one at a time l Next: Discuss probabilities associated with multiple stage processes Page 30

COMP 170 L2 Independent Trial Process Outcome at stage i independent of outcomes at earlier stages Page 31

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COMP 170 L2 Recap: l Probability starts with a process (test, experiment) whose outcome is uncertain l Sample space: the set of all possible outcomes n Might not be unique l Consider the probability of an event E n At the beginning, we have some assessment: P(E) n Then we get a new piece of information:  “Event F occurred” n Afterwards, we talk about the probability of E given F Page 39

COMP 170 L2 Recap: l If P(E|F) = P(E) n The information that “Event F occurred” does not change the probability of E n So, E is independent of F

COMP 170 L2 Recap: Outcome at stage i independent of outcomes at earlier stages