2 A large hotel has 90 bedrooms. Sometimes a guest makes a booking for a room, but then does not arrive. This is called a 'no-show'. On average \(10 \%\) of bookings are no-shows. The hotel manager accepts up to 94 bookings before saying that the hotel is full. If at least 4 of these bookings are no-shows then there will be enough rooms for all of the guests. 94 bookings have been made for each night in August. You should assume that all bookings are independent.

1. State the distribution of the number of no-shows on one night in August.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. Use a Poisson approximating distribution to find the probability that, on one night in August,
   (A) there are exactly 4 no-shows,
   (B) there are enough rooms for all of the guests who do arrive.
4. Find the probability that, on all of the 31 nights in August, there are enough rooms for all of the guests who arrive.
5. (A) In August there are \(31 \times 94 = 2914\) bookings altogether. State the exact distribution of the total number of no-shows during August.
   (B) Use a suitable approximating distribution to find the probability that there are at most 300 no-shows altogether during August.