This problem is about finding the probability that at least one of the twelve randomly selected people will have a PIN code with repeating digits.

Given:

• The probability of a randomly selected PIN code having repeating digits is 17.8%, or 0.178.
• The number of people (PIN codes) selected is 12.

To solve this, we can use the complementary probability approach. The probability that a PIN code does not have repeating digits is: $P(\text{no repeating digits}) = 1 - 0.178 = 0.822$

Now, the probability that none of the 12 people have repeating digits is: $P(\text{no repeating digits for 12 people}) = 0.822^{12}$

The probability that at least one person has repeating digits is the complement of this: $P(\text{at least one with repeating digits}) = 1 - 0.822^{12}$

Let's calculate this probability. Calculate the probability using the formula no_repeating_digits_probability = 0.822 num_people = 12

Probability that none of the 12 people have repeating digits

prob_no_repeating_digits = no_repeating_digits_probability ** num_people

Probability that at least one person has repeating digits

prob_at_least_one_repeating = 1 - prob_no_repeating_digits round(prob_at_least_one_repeating, 4)The probability that at least one of the twelve people will have a PIN code with repeating digits is approximately 0.9048 (rounded to four decimal places).

Would you like to dive deeper into any part of this calculation? Here are some related questions:

1. How does the complementary probability method work in this context?
2. What happens to the probability if we change the number of people selected?
3. How would the calculation differ if the probability of repeating digits were higher or lower?
4. Can this approach be applied to other types of probabilistic problems?
5. What are the limitations of using this method in real-world scenarios?

Tip: When dealing with probability problems like this, using the complement rule can simplify calculations, especially when you're interested in the likelihood of at least one event occurring.