Question

A farmer raises corn. Each year that he has a successful crop he grosses $17000 on expenses of $6000 for seed and labor. Sometimes his crop fails and he grosses only $8000. Each year the farmer has a chance of using two types of fertilizers: Type A at a cost of $2000 guarantees that there is a 60% chance of having a successful crop the next year, and type B at a cost of $3000 guarantees that there is an 80% chance of a good crop the next year. Determine when the farmer should use fertilizer A and B.

Answer 1

Since we have been given the probabilities we can compare the success by looking at the expectation. whichever gives a better expectation of profit.

We know that the expense of seed and labor is certain but the cost of the fertilizer and the grossing due to the use of the fertilizer are dependent on the probability of the success rate of fertilizer .

The expected profit = P(success) * Profit if successful + P(failure) * Profit if failed - Seed and Labor cost

= P(success) * (Grossing - Cost) if successful + P(failure) * (Grossing - Cost)if failed - Seed and Labor cost

A

Seed and labor = 6000

Cost = 2000

Grossing if successful = 17000

Grossing if failed = 8000   

P(success) = 60% = 0.60

Therefore P(Failure) = 1 - 0.60 =0.40

Expected profit of A = (17000 - 2000) * 0.60 + (8000 - 2000) * 0.40 - 6000

  

B

Seed and labor = 6000

Cost = 3000

Grossing if successful = 17000

Grossing if failed = 8000   

P(success) = 80% = 0.80

Therefore P(Failure) = 1 - 0.80 =0.20

Expected profit of A = (17000 - 3000) * 0.80 + (8000 - 3000) * 0.20 - 6000

  

Although the cost of the fertilizer would be certain too so probabiities would need to be only multiplied by the profits.

Since the expected profit of B fertilizer is more than A

Farmer should use fertilizer B