The juror pool for an upcoming murder trial of a celebrity...

The juror pool for an upcoming murder trial of a celebrity actor contains the names of 100,000 individuals in the population who may be called for jury duty. The percentage of available jurors on the population list who are Hispanic is 40%. A jury of 12 is selected at random from the population list of available jurors. Let x= the number of Hispanics selected to be jurors for this jury. a. Find the probability that only 1 Hispanic is selected for the jury. b. If only 1 Hispanic juror is selected out of a sample of size 12, does this cast doubt on whether the sampling was truly random? Explain.

Transcript

00:01 12 jurors are selected, supposedly at random, from a population that is 45 % in spec.

00:07 So we have 12 independent trials here.

00:12 Is this person going to be hispanic or not? the probability they are hispanic should be 45 % if this is truly completely random.

00:22 And it says the population size there.

00:25 The only thing really interesting to us about the population size is it is much larger.

00:30 Than our sample size.

00:32 So we can say that these are chosen independently.

00:37 These are independent trials because the effect of removing a person when you select them has so little impact on this size population.

00:48 So we have 12 independent trials, two outcomes for each, hispanic juror or not, and the same probability for each.

00:56 So we have the binomial distribution.

01:00 And for part a, what proportion of the jury is hispanic? so two of them out of 12 are hispanic.

01:11 So 2 out of 12 is 1 in 6, which is 0 .167.

01:20 Part b.

01:22 If we select 12 jurors completely at random, because it's the probability 2 or fewer are going to be hispanic.

01:30 So we want the probability that x the number of hispanic jurors on this new completely random jury.

01:37 Is less than or equal to 2.

01:40 Now i'm going to use the binomial formula for this.

01:43 Probability of x, hispanic jurors, is n choose x, p to be x, or minus p to be n minus x.

01:52 So this gives us the probability of an exact number.

01:55 So we're going to find probability none of them are, one of them is, and two of them are, and add these together.

02:06 We'll start with two to demonstrate.

02:07 So we have these three terms.

02:10 This term is for the two who are hispanic.

02:13 Probability of 0 .5 for each.

02:15 And because these are independent trials, we can just multiply to combine these.

02:20 So we square it.

02:22 This term is for the 10 who are not...

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