A horticulturalist knows from experience that when taking cuttings from bay trees only 15 in every 100 successfully take root.

(a) In a batch of ten randomly selected cuttings, find the probability that (i) none of the cuttings take root, (ii) fewer than three of the cuttings take root.

(b) Let n be the smallest number of cuttings which need to be examined before there is at least a 95% chance that one or more of them will have taken root. (i) Show that satisfies (0.85)" <0.05. (ii) Given that (0.85)7-0.0631, find the value of x.

(c) Using a suitable approximation, estimate the probability that fewer than six in a batch of 50 cuttings take root. (L)