Based on a poll, among adults who regret getting tattoos, $18 \%$ say that they were too young when they got their tattoos. Assume that four adults who regret getting tattoos are randomly selected, and find the indicated probability. Complete parts (a) through (d) below.
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.)
c. Find the probability that the number of selected adults saying they were too young is 0 or 1 .
(Round to four decimal places as needed.)
d. If we randomly select four adults, is 1 a significantly low number who say that they were too young to get tattoos?
, because the probability that of the selected adults say that they were too young is 0.05 .
Answer
Finally, we need to determine if 1 is a significantly low number who say that they were too young to get tattoos. Since the probability that exactly one of the selected adults says that he or she was too young to get tattoos is 0.3955, which is greater than 0.05, we can conclude that 1 is not a significantly low number.
Step 1 ：First, we need to understand the problem. We are given that 18% of adults who regret getting tattoos say that they were too young when they got their tattoos. We are asked to find the probability that exactly one of the four randomly selected adults says that he or she was too young to get tattoos.
Step 2 ：We can use the binomial probability formula to solve this problem. The binomial probability formula is \(P(X=k) = C(n, k) \cdot (p^k) \cdot ((1-p)^{n-k})\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success, and \(C(n, k)\) is the combination of \(n\) items taken \(k\) at a time.
Step 3 ：In this case, \(n=4\), \(k=1\), and \(p=0.18\). So, we can substitute these values into the formula to get the probability.
Step 4 ：The combination \(C(n, k)\) can be calculated as \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n!\) is the factorial of \(n\), \(k!\) is the factorial of \(k\), and \((n-k)!\) is the factorial of \((n-k)\).
Step 5 ：So, \(C(4, 1) = \frac{4!}{1!(4-1)!} = 4\).
Step 6 ：Substitute \(C(4, 1)\), \(n=4\), \(k=1\), and \(p=0.18\) into the binomial probability formula, we get \(P(X=1) = 4 \cdot (0.18^1) \cdot ((1-0.18)^{4-1})\).
Step 7 ：Calculate the above expression to get the probability that exactly one of the selected adults says that he or she was too young to get tattoos. The result is approximately 0.3955.
Step 8 ：Next, we need to find the probability that the number of selected adults saying they were too young is 0 or 1. This is the sum of the probabilities that exactly 0 or 1 of the selected adults says that he or she was too young to get tattoos.
Step 9 ：First, calculate the probability that exactly 0 of the selected adults says that he or she was too young to get tattoos. Using the binomial probability formula, we get \(P(X=0) = C(4, 0) \cdot (0.18^0) \cdot ((1-0.18)^{4-0})\).
Step 10 ：Calculate the above expression to get the probability that exactly 0 of the selected adults says that he or she was too young to get tattoos. The result is approximately 0.4494.
Step 11 ：Then, add the probabilities that exactly 0 or 1 of the selected adults says that he or she was too young to get tattoos. The result is approximately 0.8449.
Step 12 ：Finally, we need to determine if 1 is a significantly low number who say that they were too young to get tattoos. Since the probability that exactly one of the selected adults says that he or she was too young to get tattoos is 0.3955, which is greater than 0.05, we can conclude that 1 is not a significantly low number.
