6. A farmer wants to plant one of three crops: wheat, corn, or soybeans. The amount of rain received will determine how well or poor the crop does. The payoffs are:

Decision State of Nature
Crop Abundant Rain (a) Marginal Rain (m)
Wheat $5,000$3,000
Corn $10,000$-4,000
Soybeans $3,000$1,000

Given:
(1) Prior data: P(a) = .60 and P(m) = .40

The farmer considers hiring a weather forecaster to get additional understanding of the expected weather for the year. The forecaster will report the weather will be either wet or dry.
a = Abundant rain
m = Marginal rain
W = Wet weather forecast
D = Dry weather forecast

Conditional Probabilities

P (W|a) = .80
P (D|a) = .20
P (W|m) = .10
P (D|m) = .90

Questions:
(1) Draw a decision tree for the overall situation.

(2) Calculate the Posterior Probabilities of (D) and P(W).
(3) How much should he consider paying the weather forecaster? What is the maximum for this payment?
(4) Determine the efficiency of the additional information?
(5) What is the farmer’s decision strategy?

2. Originally Posted by lillian
6. A farmer wants to plant one of three crops: wheat, corn, or soybeans. The amount of rain received will determine how well or poor the crop does. The payoffs are:

Decision State of Nature
Crop Abundant Rain (a) Marginal Rain (m)
Wheat $5,000$3,000
Corn $10,000$-4,000
Soybeans $3,000$1,000

Given:
(1) Prior data: P(a) = .60 and P(m) = .40

The farmer considers hiring a weather forecaster to get additional understanding of the expected weather for the year. The forecaster will report the weather will be either wet or dry.
a = Abundant rain
m = Marginal rain
W = Wet weather forecast
D = Dry weather forecast

Conditional Probabilities

P (W|a) = .80
P (D|a) = .20
P (W|m) = .10
P (D|m) = .90

Questions:
(1) Draw a decision tree for the overall situation.
Decision tree attached

RonL

3. Originally Posted by lillian
6. A farmer wants to plant one of three crops: wheat, corn, or soybeans. The amount of rain received will determine how well or poor the crop does. The payoffs are:

Decision State of Nature
Crop Abundant Rain (a) Marginal Rain (m)
Wheat $5,000$3,000
Corn $10,000$-4,000
Soybeans $3,000$1,000

Given:
(1) Prior data: P(a) = .60 and P(m) = .40

The farmer considers hiring a weather forecaster to get additional understanding of the expected weather for the year. The forecaster will report the weather will be either wet or dry.
a = Abundant rain
m = Marginal rain
W = Wet weather forecast
D = Dry weather forecast

Conditional Probabilities

P (W|a) = .80
P (D|a) = .20
P (W|m) = .10
P (D|m) = .90

Questions:
(1) Draw a decision tree for the overall situation.

(2) Calculate the Posterior Probabilities of (D) and P(W).
$\displaystyle P(W)= P(W|a)p(a) + P(W|m)p(m) = 0.8\times 0.6 + 0.1 \times 0.4=0.52$

$\displaystyle P(D)= P(D|a)p(a) + P(D|m)p(m) = 0.2\times 0.6 + 0.9 \times 0.4=0.48$

RonL

4. Originally Posted by lillian
6. A farmer wants to plant one of three crops: wheat, corn, or soybeans. The amount of rain received will determine how well or poor the crop does. The payoffs are:

Decision State of Nature
Crop Abundant Rain (a) Marginal Rain (m)
Wheat $5,000$3,000
Corn $10,000$-4,000
Soybeans $3,000$1,000

Given:
(1) Prior data: P(a) = .60 and P(m) = .40

The farmer considers hiring a weather forecaster to get additional understanding of the expected weather for the year. The forecaster will report the weather will be either wet or dry.
a = Abundant rain
m = Marginal rain
W = Wet weather forecast
D = Dry weather forecast

Conditional Probabilities

P (W|a) = .80
P (D|a) = .20
P (W|m) = .10
P (D|m) = .90

Questions:
(1) Draw a decision tree for the overall situation.

(2) Calculate the Posterior Probabilities of (D) and P(W).
(3) How much should he consider paying the weather forecaster? What is the maximum for this payment?
The maximum he should pay is the difference between the expected return
if he plants optimally with the forcast and the expected return if he plants
optimally without the forcast.

However I would be tempted to pay less.

RonL

:d