Question

A county is 50% African-American and 50% Caucasian, and an African-American is on trial. A jury has been selected that contains six Caucasians and two African-Americans (this is an 8 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 three methods to gain evidence to advise the man on trial:
Method 1 (optional ): Calculate a theoretical probability by building a tree diagram that displays all possible jury make-ups
Method 2: Calculate a theoretical probability by using the binomial formula
Method 3: Perform a simulation in Minitab and compare results
with the "true" probabilities compute in Method 2. 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 you decision.

Answer 1

The defendant claim is credible. It should be advised that the defendant can pursue his claim of racial bias in jury

Explanation:

Since the county comprises of 50% African-American and 50% Caucasian. If a jury of 8 members are to be selected from the county's people,

The probability of selection of 4 African-American is 4/8(50%) = 1/2

The probability of selection of 4 Caucasian is 4/8(50%) = 1/2.

This above is a binomial probability distributions. It is a probability of two possible outcomes, summing up to equals 1.

Selection of 2 African-American and 6 Caucasian in 8 jury members actually show a racial bias. It does not reflect 50% each proportion and probability.

Probability of African-American Jury selection is 2/8(1/4)

Probability of Caucasian Jury selection is 6/8(3/4)