Question

In a unique grove of 100 trees, a company is considering bioprospecting for a cure for a serious disease. Assume each tree has an equal independent probability p, 0 < p <1, of containing a substance that will lead to a cure, and such a substance is worth $5 Billion to the company. There are no other trees in existence with this property. Bioprospecting cost is zero per tree.

(i) Write and explain an expression for the expected value of a grove of 99 trees.

(ii) Write and explain an expression for the marginal expected value of having a tree in the grove (that is, the expected extra value of a grove of 100 trees rather than 99) to the company.

Answer 1

So, here “p” be the probability of containing a subsistence that leads to a cure of each tree, => “1-p”, be the probability of not containing the subsistence. Now, the company will be able to make the cure of the disease if at least one tree have the subsistence. So, the probability that at least one tree out of 99 trees have the subsistence is given below.

=> P(X > 0) = (1-p)^99.

=> the expected numbers of trees in the grove is given by, “99*P(X > 0) = 99*(1-p)^99.

=> the expected value of a grove of 99 trees is given by, “$5*99*(1-p)^99 billion”.

ii).

So, the “expected value of the of grove of 100 trees” is given by, “$5*100*(1-p)^100 billion” in tghe same way we can derive this expression. So, the marginal expected value of having a tree in the grove is given by the difference between the above two expression.

=> $5*100*(1-p)^100 - $5*99*(1-p)^99, => $5*(1-p)^99*[100*(1-p) – 99]

= $5*(1-p)^99*[100 – 100*p – 99] = $5*(1-p)^99*(1– 100*p).