In a recent survey, \( 62 \% \) of the community favored building a police substation in their neighborhood. If 14 citizens are chosen, find the binomial probability that exactly 11 of them favor the building of the police substation. 0.002 0.104 0.786 0.347 0.620

To find the binomial probability that exactly 11 out of 14 citizens favor the building of the police substation, we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} \] where: - \( n \) is the total number of trials (14 citizens), - \( k \) is the number of successful trials (11 citizens favoring the police substation), - \( p \) is the probability of success (62% or 0.62) in a single trial. Substitute the given values into the formula: \[ P(X = 11) = \binom{14}{11} \times 0.62^{11} \times (1-0.62)^{14-11} \] Now, we can calculate the binomial probability using the formula. Calculate the value by following steps: - step0: Calculate: \( { }_{14}C_{11}\times 0.62^{11}\left(1-0.62\right)^{14-11}\) - step1: Subtract the numbers: \( { }_{14}C_{11}\times 0.62^{11}\times 0.38^{14-11}\) - step2: Subtract the numbers: \( { }_{14}C_{11}\times 0.62^{11}\times 0.38^{3}\) - step3: Expand the expression: \(\frac{14!}{11!\times \left(14-11\right)!}\times 0.62^{11}\times 0.38^{3}\) - step4: Subtract the numbers: \(\frac{14!}{11!\times 3!}\times 0.62^{11}\times 0.38^{3}\) - step5: Reduce the fraction: \(\frac{14\times 13\times 12}{3!}\times 0.62^{11}\times 0.38^{3}\) - step6: Multiply the terms: \(\frac{2184}{3!}\times 0.62^{11}\times 0.38^{3}\) - step7: Divide the terms: \(364\times 0.62^{11}\times 0.38^{3}\) - step8: Convert the expressions: \(364\left(\frac{31}{50}\right)^{11}\times 0.38^{3}\) - step9: Convert the expressions: \(364\left(\frac{31}{50}\right)^{11}\left(\frac{19}{50}\right)^{3}\) - step10: Simplify: \(364\times \frac{31^{11}}{50^{11}}\times \left(\frac{19}{50}\right)^{3}\) - step11: Multiply the terms: \(\frac{91\times 31^{11}}{512\times 25^{11}}\times \left(\frac{19}{50}\right)^{3}\) - step12: Evaluate the power: \(\frac{91\times 31^{11}}{512\times 25^{11}}\times \frac{19^{3}}{50^{3}}\) - step13: Multiply the fractions: \(\frac{91\times 31^{11}\times 19^{3}}{512\times 25^{11}\times 50^{3}}\) - step14: Multiply: \(\frac{624169\times 31^{11}}{512\times 25^{11}\times 50^{3}}\) The binomial probability that exactly 11 out of 14 citizens favor the building of the police substation is approximately 0.1039 or 10.394%. Therefore, the correct answer is 0.104.