Answer to Question #323835 in Statistics and Probability for Bless

Question

 A biologist needs at least 3 mature specimens of a certain plant. The plant needs a year to reach maturity; once a seed is planted, any plant will survive for the year with probability 1/1000 (independently of other plants). The biologist plants 3000 seeds. A year is deemed a success if three or more plants from these seeds reach maturity.

(a) Write down the exact expression for the probability that the biologist will indeed end up with at least 3 mature plants.

(b) Write down a relevant approximate expression for the probability from (a). Justify briefly the approximation.

(c) The biologist plans to do this year after year. What is the approximate probability that he has at least 2 successes in 10 years?

(d) Devise a method to determine the number of seeds the biologist should plant in order to get at least 3 mature plants in a year with probability at least 0:999. (Your method will probably require a lengthy calculation (do not try to carry it out with pen and paper.)