Question: Enter your PIN: The technology company DataGenetics suggests that 17.8% of all four-digit personal…

Enter your PIN: The technology company DataGenetics suggests that 17.8% of all four-digit personal identification numbers, or PIN codes, have a repeating digits format such as 2525. Assuming this to be true, if the PIN codes of twelve people are selected at random, what is the probability that at least one of them will have repeating digits? Round your answer to four decimal places.

The probability that at least one of the codes will have repeating digits is \(\boxed{\phantom{0.0000}}\).

Solution

To solve this problem, we need to find the probability that at least one of the selected PIN codes has repeating digits. We are given that 17.8% of all four-digit PIN codes have repeating digits. We will use complementary probability to solve this. First, calculate the probability that a randomly selected PIN code does not have repeating digits. \[ P(\text{no repeats}) = 1 - 0.178 = 0.822 \] Next, calculate the probability that all twelve selected PIN codes do not have repeating digits. \[ P(\text{all 12 no repeats}) = (0.822)^{12} \] Now, calculate the probability that at least one of the twelve PIN codes has repeating digits. \[ P(\text{at least 1 repeat}) = 1 - P(\text{all 12 no repeats}) = 1 - (0.822)^{12} \] Calculate the final probability. \[ P(\text{at least 1 repeat}) \approx 1 - 0.08276 = 0.9172 \] Thus, the probability that at least one of the codes will have repeating digits is 0.9172.