The florist section in a grocery store stocks 18 dozen roses at the beginning of each week. On the average, the florist sells 3 dozens a day (one dozen at a time), but the actual demand follows a Poisson distribution. Whenever the stock level reaches 5 dozens, a new order of 18 new dozens is placed for delivery at the beginning of the following week. Because of the nature of the item, all roses left at the end of the week are disposed of. Determine the following: *(a) The probability that the stock is depleted after 3 days.

(b) The average number of dozen roses left at the end of the second day. *

(c) The probability that at least one dozen is purchased by the end of the fourth day, given that the last dozen was bought at the end of the third day.

(d) The probability that the time remaining until the next purchase is at most half a day, given that the last purchase occurred a day earlier.

(e) The probability that no purchases will occur during the first day.

(f) The probability that no order will be placed by the end of the week.