Probability - Waiting-Time and Stopping-Rule Probabilities

Customers wait in a queue in a supermarket. A customer can choose to pay with cash or a credit card. Assume that whether or not a customer pays with a credit card is independent of any other customers’ methods of payment. It is known that \(60\%\) of customers choose to pay with a credit card.

1. Find the probability that the first three customers pay with a credit card and the next three pay with cash;
2. Find the probability that the \(4^{\mathrm{th}}\) person who pays with credit card is the \(6^{\mathrm{th}}\) customer in the queue.
3. There are \(15\) customers in the queue. Find the probability that there are more than \(10\) customers who pay with a credit card and the \(4^{\mathrm{th}}\) person who pays with credit card is the \(6^{\mathrm{th}}\) customer in the queue.

There are \(n\) customers waiting in another queue in the same supermarket. The probability that at least one customer pays with cash is greater than \(0.995\).

Find the minimum value of \(n\).