A county is 40% African-American and 60% Caucasian, and an African-American is on trial. A jury has been selected that contains eight Caucasians and two African-Americans (this is a 10 person jury). The defendant claims racial bias in the jury selection because its makeup seems unlikely given the racial percentages in the county. However, the prosecutor claims that the jury was selected without regard to race. Is the defendant's claim credible? In other words, is selecting 2 or fewer African-Americans unlikely to happen by chance? The defendant needs to know these probabilities in order to see if he should pursue his claim. If he pursues this claim and is found correct, he could get a new jury and a better chance at acquittal. However, if he is found incorrect he will have spent more money for his lawyer to pursue the claim and will have the same jury that has been chosen.

The defendant has hired your investigative team to investigate this claim. By your vast statistical expertise you have decided to employ two methods to gain evidence to advise the man on trial:

Method 1: Calculate a theoretical probability by using the binomial formula

Method 2: Perform a computer simulation and compare results with the "true" probabilities computed in Method 1.

You will compare these probabilities to each other, make a decision, and advise the man.

The question you need to answer:

Should the defendant pursue his claim of racial bias in jury selection or should he stick with the jury that was selected because most likely it just happened by chance? Make sure that you provide a solid statistical support for your decision, please include in your report results for the actual probabilities and the probabilities obtained from the simulation.

Hint: If the probability of having two or less African-Americans in a jury of 10 people is greater than 0.05 (5%), this is not considered a rare event and the defendant's claim should not be pursued, please include the comparison of these probabilities as the basis to support your conclusion.

Please include tables listing the exact probabilities and a table summarizing the relative frequencies of your simulated probabilities in your report