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ContinuousDist-4

ContinuousDist-4 - Balls Urns and the Supreme Court Supreme...

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1 Balls, Urns, and the Supreme Court Supreme Court case: Berghuis v. Smith If a group is underrepresented in a jury pool, how do you tell? Article by Erin Miller Friday, January 22, 2010 Thanks to Josh Falk for pointing out this article Justice Breyer [Stanford Alum] opened the questioning by invoking the binomial theorem. He hypothesized a scenario involving “an urn with a thousand balls, and sixty are red, and nine hundred forty are black, and then you select them at random… twelve at a time.” According to Justice Breyer and the binomial theorem, if the red balls were black jurors then “you would expect… something like a third to a half of juries would have at least one black person” on them. Justice Scalia’s rejoinder: “We don’t have any urns here.” Justice Breyer Meets CS109 Should model this combinatorially (X ~ HypGeo) Ball draws not independent trials (balls not replaced) Exact solution: P(draw 12 black balls) = 0.4739 P(draw ≥ 1 red ball) = 1 – P(draw 12 black balls) 0.5261 Approximation using Binomial distribution Assume P(red ball) constant for every draw = 60/1000 X = # red balls drawn. X ~ Bin(12, 60/1000 = 0.06) P(X ≥ 1 ) = 1 P(X = 0) 1 0.4759 = 0.5240 In Breyer’s description, should actually expect just over half of juries to have at least one black person on them 12 1000 12 940 Demo From Discrete to Continuous So far, all random variables we saw were discrete Have finite or countably infinite values (e.g., integers) Usually, values are binary or represent a count Now it’s time for continuous random variables Have (uncountably) infinite values (e.g., real numbers)

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