The number of oil tankers arriving at a port between successive high tides has a Poisson distribution with mean 2. The depth of the water is such that loaded vessels can enter the dock area only on the high tide. The port has dock space for only three tankers, which are discharged and leave the dock area before the next tide. Only the first three loaded tankers waiting at any high tide go into the dock area; any others must await another high tide. Starting from an evening high tide after which no ships remain waiting their turn, find (to three decimal places) the probabilities that after the next morning's high tide

(a) the three dock berths remain empty,

(b) the three berths are all filled. Find (to two decimal places) the probability that no tankers are left waiting outside the dock area after the following evening's high tide. (NEAB)