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Unformatted text preview: 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)Have (uncountably) infinite values (e....
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This document was uploaded on 12/24/2011.
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