A boutique hotel is managing reservations for a given night. Currently, it has 15 rooms available for reservation. When a customer makes a reservation, there is a 0.30 chance of the customer cancelling a reservation, and a 0.70 chance of the customer showing for the reservation. Each customer that shows up pays a rate of $300, while each customer that cancels is charged a $100 cancellation fee. Suppose that the hotel takes exactly 15 reservations. Dismayed by the high rate of cancellations, the hotel considers the possibility of overbooking, i. e., accepting more reservations than the available capacity of 15 rooms.
What is the largest number of reservations the hotel should accept to ensure that the probability of having insufficient capacity (i. e., the number of customers who show up is higher than the number of available rooms) is at most 0.25?