Based on a poll, among adults who regret getting tattoos, \( 20 \% \) say that they were too young when they got their tattoos. Assume that nine adults who regret getting tattoos are randomly selected, and find the indicated probability. Complete parts (a) through (d) below. a. Find the probability that none of the selected adults say that they were too young to get tattoos. b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos. (Round to four decimal places as needed.)

Given: - Among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. - Nine adults who regret getting tattoos are randomly selected. Let's denote: - \( p \) as the probability that an adult regrets getting tattoos and says they were too young when they got their tattoos. - \( q \) as the probability that an adult regrets getting tattoos but does not say they were too young when they got their tattoos. We are given that \( p = 0.20 \) and \( q = 1 - p = 0.80 \). a. To find the probability that none of the selected adults say that they were too young to get tattoos, we need to find the probability of selecting 0 adults who say they were too young when they got their tattoos out of the 9 selected adults. The probability of selecting 0 adults who say they were too young when they got their tattoos is given by the binomial probability formula: \[ P(X = k) = \binom{n}{k} \cdot p^k \cdot q^{n-k} \] Where: - \( n = 9 \) (number of adults selected) - \( k = 0 \) (number of adults who say they were too young) - \( p = 0.20 \) - \( q = 0.80 \) Substitute the values into the formula to find the probability. Calculate the value by following steps: - step0: Calculate: \( { }_{9}C_{0}\times 0.2^{0}\times 0.8^{9}\) - step1: Evaluate the power: \( { }_{9}C_{0}\times 1\times 0.8^{9}\) - step2: Expand the expression: \(\frac{9!}{0!\times \left(9-0\right)!}\times 1\times 0.8^{9}\) - step3: Remove 0: \(\frac{9!}{0!\times 9!}\times 1\times 0.8^{9}\) - step4: Reduce the fraction: \(\frac{9!}{9!}\times 1\times 0.8^{9}\) - step5: Divide the terms: \(1\times 1\times 0.8^{9}\) - step6: Convert the expressions: \(1\times 1\times \left(\frac{4}{5}\right)^{9}\) - step7: Multiply the terms: \(\left(\frac{4}{5}\right)^{9}\) - step8: Use the properties of exponents: \(\frac{4^{9}}{5^{9}}\) The probability that none of the selected adults say that they were too young to get tattoos is approximately 0.1342. b. To find the probability that exactly one of the selected adults says that he or she was too young to get tattoos, we need to find the probability of selecting 1 adult who says they were too young when they got their tattoos out of the 9 selected adults. The probability of selecting exactly 1 adult who says they were too young when they got their tattoos is also given by the binomial probability formula: \[ P(X = k) = \binom{n}{k} \cdot p^k \cdot q^{n-k} \] Where: - \( n = 9 \) (number of adults selected) - \( k = 1 \) (number of adults who say they were too young) - \( p = 0.20 \) - \( q = 0.80 \) Substitute the values into the formula to find the probability. Calculate the value by following steps: - step0: Calculate: \( { }_{9}C_{1}\times 0.2^{1}\times 0.8^{8}\) - step1: Expand the expression: \(\frac{9!}{1!\times \left(9-1\right)!}\times 0.2^{1}\times 0.8^{8}\) - step2: Subtract the numbers: \(\frac{9!}{1!\times 8!}\times 0.2^{1}\times 0.8^{8}\) - step3: Reduce the fraction: \(\frac{9!}{8!}\times 0.2^{1}\times 0.8^{8}\) - step4: Reduce the fraction: \(9\times 0.2^{1}\times 0.8^{8}\) - step5: Calculate: \(9\times \frac{1}{5}\times 0.8^{8}\) - step6: Convert the expressions: \(9\times \frac{1}{5}\left(\frac{4}{5}\right)^{8}\) - step7: Multiply the terms: \(\frac{9}{5}\left(\frac{4}{5}\right)^{8}\) - step8: Evaluate the power: \(\frac{9}{5}\times \frac{4^{8}}{5^{8}}\) - step9: Multiply the fractions: \(\frac{9\times 4^{8}}{5\times 5^{8}}\) - step10: Multiply: \(\frac{9\times 4^{8}}{5^{9}}\) - step11: Calculate: \(\frac{589824}{5^{9}}\) The probability that exactly one of the selected adults says that he or she was too young to get tattoos is approximately 0.3020.