A Gallup poll of 1236 adults showed that $12 \%$ of the respondents believe that it is bad luck to walk under a ladder. Consider the probability that among 30 randomly selected people from the 1236 who were polled, there are at least 2 who have that belief. Given that the subjects surveyed were selected without replacement, the events are not independent. Can the probabiity be found by using the binomial probability formula? Why or why not?
Choose the correct answer below.
A. Yes. There are a fixed number of selections that are independent, can be classified into two categories, and the probability of success remains the same.
B. No. The selections are not independent, and the $5 \%$ guideline is not met.
C. Yes. Athough the selections are not independent, they can be treated as being independent by applying the $5 \%$ guideline.
D. No. The selections are not independent.
Answer
\(\boxed{\text{The correct answer is B. No. The selections are not independent, and the $5 \%$ guideline is not met.}}\)
Step 1 ：The binomial probability formula can be used when the events are independent, which means the outcome of one event does not affect the outcome of another event.
Step 2 ：In this case, the subjects are selected without replacement, which means the events are not independent.
Step 3 ：The $5 \%$ guideline is a rule of thumb that says if you are sampling without replacement and your sample size is less than $5 \%$ of the population size, you can treat the selections as being independent.
Step 4 ：In this case, the sample size (30) is more than $5 \%$ of the population size (1236), so the $5 \%$ guideline is not met.
Step 5 ：Therefore, the binomial probability formula cannot be used.
Step 6 ：\(\boxed{\text{The correct answer is B. No. The selections are not independent, and the $5 \%$ guideline is not met.}}\)
