The probability that a cashier makes a mistake on any transaction is 0.05 . A cashier makes 3 transactions in a 10-minute period. Calculate the following values. Write answers for this problem with 6 decimal places.
1. The probability that all 3 transactions had errors.
2. The probability that none of the 3 transactions had errors.
3. The probability that at least one transaction had an error.

Step 1 ：This problem is about calculating probabilities using the binomial distribution model. The binomial distribution model is used to find the probability of success of an event which has only two possible outcomes in a series of experiments. If a random variable X follows binomial distribution with parameters n and p, we write X ~ B(n, p). Here, n is the number of trials and p is the probability of success in a single trial.

Step 2 ：In this problem, a cashier makes 3 transactions in a 10-minute period. The probability that the cashier makes a mistake on any transaction is 0.05. So, n = 3 and p = 0.05.

Step 3 ：We are asked to calculate the following probabilities:

Step 4 ：The probability that all 3 transactions had errors. This is the probability of 3 successes (errors in this case) in 3 trials. We can use the formula for binomial probability to calculate this. The calculated probability is \(0.000125\).

Step 5 ：The probability that none of the 3 transactions had errors. This is the probability of 0 successes in 3 trials. Again, we can use the binomial probability formula. The calculated probability is \(0.857375\).

Step 6 ：The probability that at least one transaction had an error. This is 1 minus the probability that none of the transactions had errors. The calculated probability is \(0.142625\).

Step 7 ：Final Answer: \(\boxed{0.000125}\) is the probability that all 3 transactions had errors, \(\boxed{0.857375}\) is the probability that none of the 3 transactions had errors, and \(\boxed{0.142625}\) is the probability that at least one transaction had an error.