I go through the following process Close my eyes and pick a card Pick a side at

# I go through the following process close my eyes and

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I go through the following process. Close my eyes and pick a card Pick a side at random Show you that side Suppose I show you red. What is the probability the other side is red too? Let X 1 be the random side of the random card I chose Let X 2 be the other side of that card Compute P ( X 2 = red | X 1 = red ) P ( X 2 = red | X 1 = red ) = P ( X 1 = R , X 2 = R ) P ( X 1 = R ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 14 / 69

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Now we can solve the card problem Now we can solve the card problem. Let X 1 be the random side of the random card I chose Let X 2 be the other side of that card Compute P ( X 2 = red | X 1 = red ) P ( X 2 = red | X 1 = red ) = P ( X 1 = R , X 2 = R ) P ( X 1 = R ) Numerator is 1/3: Only one card has two red sides. Denominator is 1/2: Three sides out of six are red. So P ( X 2 = red | X 1 = red ) = 2 / 3 Is it possible for a conditional probability to be outside of [0 , 1]? COS 424/SML 302 Probability and Statistics Review February 6, 2019 15 / 69
Gender bias at Berkeley: Conditional probabilities From the textbook Statistics by Freedman, Pisani, & Purves: These numbers show that the system is biased ( p = 1 . 333 × 10 - 10 ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 16 / 69

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Gender bias at Berkeley: Simpson’s Paradox From the textbook Statistics by Freedman, Pisani, & Purves: When the graduate school asked each of the departments to make changes in their admissions policy, all of the departments noted that none of their admissions numbers were biased against women. What important variable are we not incorporating in this comparison? COS 424/SML 302 Probability and Statistics Review February 6, 2019 17 / 69
Gender bias at Berkeley: Condition on department From the textbook Statistics by Freedman, Pisani, & Purves: Conditioning on the department, the bias is reversed. Why? COS 424/SML 302 Probability and Statistics Review February 6, 2019 18 / 69

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Gender bias at Berkeley: Condition on department From the textbook Statistics by Freedman, Pisani, & Purves: Women are more likely than men to apply to departments with low admission rates. COS 424/SML 302 Probability and Statistics Review February 6, 2019 19 / 69
The chain rule The definition of conditional probability lets us derive the chain rule The chain rule factorizes a joint distribution as a product of conditional distributions: P ( X , Y ) = P ( X , Y ) P ( Y ) P ( Y ) = P ( X | Y ) P ( Y ) Example: use conditional and marginal to get joint distribution Let Y be a disease and X be a symptom. We may know P ( X | Y ) and P ( Y ) from data. Use chain rule to find the probability of the disease and the symptom. COS 424/SML 302 Probability and Statistics Review February 6, 2019 20 / 69

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The chain rule across n variables In general, for any set of n variables P ( X 1 , . . . , X n ) = P ( X 1 ) n Y i =2 P ( X i | X 1 , . . . , X i - 1 ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 21 / 69
Marginalization Given a collection of RVs, we might only consider a subset of them.

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