New Exercise Bank Content Qu$.12-08$

Dwight Moody Is the manager of a large farm with $1,000$ acres of arable land. For greater efficlency, Dwight always devotes the farm to

growing one crop at a time. He now needs to make a decision on which one of four crops to grow during the upcoming growing

season. For each of these crops, Dwight has obtained the following estimates of crop ylelds and net Incomes per bushel under varlous

weather conditions.

Weather

Dry

Moderate

Damp

Net income per bushel

Expected Yield, Bushels$/$Acre

Crop $2$

$15$

$20$

$30$

$$1\mathrm{.}50$

After referring to historical meteorological records, Dwight also has estimated the following prior probabilities for the weather during

the growing season:

Dry

Moderate

Damp

Construct a decision tree for this problem and use Bayes' decision rule to determine which crop to grow.

a$.$ Determine which crop to grow.

Crop to grow

Crop $1$

$20$

$35$

$40$

$$1\mathrm{.}00$

Crop $3$

$30$

$25$

$25$

$$1\mathrm{.}00$

Crop $4$

$40$

$40$

$$0\mathrm{.}50$

$40$

$0\mathrm{.}3$

$0\mathrm{.}5$

$0\mathrm{.}2$

b$.$ Determine the expected payoff.

Note: Round your answer to $2$ decimal places.

Expected payoff