Please i need solutions

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.)

Answer 1

Hi Emmanuel A.,

This is indeed a probability problem, not a biology problem!

In particular, you need combinations of failed-seeds and matured-seeds.

Then, you can confidently say that the probability of getting 3 or more mature plants =

1 - [P(0 matured plants) +P(1 matured plant) + P(2 matured plants)]

So what are the combinations? They are the data on failed vs. matured plants.

so P(0 matured plants) = 0.999^3000 ~ 0.04971

P(1 matured plant) = 0.999^2999 * (0.001)^1 * (3000)!/((2999)!(1)!) ~ 0.14928

P(2 matured plants) = 0.999^2998 * (0.001)^2 * (3000)!/((2998)!2!) ~ 0.22408

You should be able to deduce the format of each calculation here, it is P(each failed)^(#failed) *P(each matured)^(#matured) * ways of getting that mix as a combination, that is =(#total)! / ((#failed)!(#matured)!)

The rest of the problem is solved similarly.

Incidentally, this is an abysmal rate of seedling success, assuming the biologist is working in a greenhouse, i.e. cold-stratifying the seeds (or whatever program they need), germinating, and cultivating. And if the biologist is putting them out in the field, in adverse conditions, that wouldn't necessarily reflect the natural process of self-seeding. AND, if the biologist is intending to preserve the species by this program, 3 plants is far too few to ensure genetic diversity preservation. Just sayin'.

-- Cheers, --Mr. d.