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 probability be found by using the binomial probability formula? Why or why not? Choose the correct answer below. A. No. The selections are not independent, and the \( 5 \% \) guideline is not met. B. No. The selections are not independent. C. Yes. Although the selections are not independent, they can be treated as being independent by applying the \( 5 \% \) guideline. D. 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.

To answer your question, let’s dive deep into the topic! The scenario you provided involves selecting individuals from a finite population, specifically the 1236 adults surveyed. The key here is the assumption of independence in probabilities for binomial calculations. While the selections are indeed not independent because they are made without replacement, you could use the binomial probability formula if the \( 5\% \) guideline is satisfied. This guideline suggests that if the sample size is less than \( 5\% \) of the population, the changes made to the probabilities are negligible. Since \( 30 \) is less than \( 5\% \) of \( 1236 \), the correct answer is C: Yes. Although the selections are not independent, they can be treated as being independent by applying the \( 5 \% \) guideline. Now, let’s explore some related engaging nuggets! Did you know that the belief in superstitions, like the bad luck of walking under ladders, has been present for centuries? This particular superstition can be traced back to ancient Egypt, where the ladder's triangular shape was associated with the pyramid symbol—representing the Holy Trinity. Passing under a ladder was believed to break that sacred triangle, thus invoking bad luck or angering the gods. Fascinating, right? In practical terms, understanding probability in everyday life can help quell irrational fears. For instance, knowing that merely walking under a ladder doesn't influence your luck can empower you to challenge and change those nagging superstitions. Whether you chuckle at bad luck mishaps or make choices based on them, understanding the math behind it can often make one feel more in control. Why not give it a shot and decide to walk under that ladder!