Fourteen jurors are randomly selected from a population of 5 million residents. Of these 5 million residents, it is known that \( 48 \% \) are of a minority race. Of the 14 jurors selected, 2 are minorities. (a) What proportion of the jury described is from a minority race? (b) If 14 jurors are randomly selected from a population where \( 48 \% \) are minorities, what is the probability that 2 or fewer jurors will be minorities? (c) What might the lawyer of a defendant from this minority race argue? (a) The proportion of the jury described that is from a minority race is 0.14 . (Round to two decimal places as needed.) (b) The probability that 2 or fewer out of 14 jurors are minorities, assuming that the proportion of the population that are minorities is \( 48 \% \), is \( \square \). (Round to four decimal places as needed.)

(a) The proportion of the jury described that is from a minority race is calculated as the number of minority jurors divided by the total number of jurors. Here, there are 2 minority jurors out of 14 total jurors, so the proportion is \( \frac{2}{14} = 0.14 \) or 14%. (b) To calculate the probability of selecting 2 or fewer minority jurors from a population where \( 48\% \) are minorities, one would typically use a binomial probability formula. The probability of exactly \( k \) successes (minorities in this case) in \( n \) trials (jurors) can be calculated using the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( p = 0.48 \), \( n = 14 \), and \( k \) can be 0, 1, or 2. Calculating for \( P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \): 1. For \( P(X = 0) \): \[ P(X = 0) = \binom{14}{0} (0.48)^0 (0.52)^{14} = 1 \times 1 \times (0.52)^{14} \approx 0.0220 \] 2. For \( P(X = 1) \): \[ P(X = 1) = \binom{14}{1} (0.48)^1 (0.52)^{13} = 14 \times 0.48 \times (0.52)^{13} \approx 0.0950 \] 3. For \( P(X = 2) \): \[ P(X = 2) = \binom{14}{2} (0.48)^2 (0.52)^{12} = 91 \times (0.48)^2 \times (0.52)^{12} \approx 0.1990 \] Now summing these probabilities: \[ P(X \leq 2) \approx 0.0220 + 0.0950 + 0.1990 = 0.3160 \] Thus, the probability that 2 or fewer out of 14 jurors are minorities is approximately 0.3160 (rounded to four decimal places as needed). (c) The lawyer representing a defendant from a minority race might argue that the underrepresentation of their racial group on the jury could lead to biased decision-making, thus potentially compromising the fairness of the trial. They may highlight that the selected jury does not reflect the demographics of the larger community, which could affect the jury's ability to empathize with the defendant's experiences and challenges faced by their racial group, thereby emphasizing the need for a more inclusive jury selection process to ensure justice.