Representative Jury Pools

In the Supreme Court case: Berghuis v. Smith, the Supreme Court (of the US) discussed the question: "If a group is underrepresented in a jury pool, how do you tell?"

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 green, 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. 

Note: What is missing in this conversation is the power of diverse backgrounds when making difficult decisions.

Explination:

You can think of the number of under-representative jurrors as being a Hyper Geometric Random Variable st

\begin{align*}X \sim HypGeo(n=12, N = 1000, m = 60)\end{align*}

\begin{align*} P(X \geq 1) &= 1 - P(X = 0) \\ &= 1 - \frac{ {60 \choose 0}{940 \choose 12} }{1000 \choose 12} \\ &\approx 0.5261 \end{align*}