A citrus grower anticipates a profit of $100,000 this year if the nightly temperatures remain mild. Unfortunately, the weather forecast indicates a 10% chance that the temperature will drop below freezing during the next week. Such freezing weather will destroy 40% of the crop and reduce the profit to $60,000. However, the grower can protect the citrus fruit against the possible freezing at a cost of $5000. Should the grower spend the $5000 and thereby reduce the profit to $95,000? Hint: Compute E(X), where X is the profit the grower will get if he does nothing to protect the fruit.

No, the citrus grower shouldn't spend the $5000 and thereby reduce the profit to $95,000 as the expected profit from doing nothing to protect the citrus plants ($96,000) is more than the profit that'll be available if $5,000 is spent on protection.

Step-by-step explanation:

First of, we compute the probability distribution of X.

X represents the profit if the citrus grower does nothing to protect the citrus fruits.

If the citrus grower does nothing, there are two possibilities as to what will happen.

1) The temperatures can drop below freezing point at a chance of 10% and the profit plummets to $60,000

2) The temperature can remain mild at a chance of 90% (100%-10%) and the profit stays at $100,000.

The probability distribution will then be

X P(X)

60,000 ||| 0.10

100,000 | 0.90

The expected value of any probability distribution is given as

E(X) = Σ xᵢpᵢ

xᵢ = each variable

pᵢ = probability of each variable

E(X) = (60,000×0.10) + (100,000×0.90)

= 6,000 + 90,000 = $96,000

The expected amount of profits from doing nothing to protect the citrus fruits = $96,000

The expected amount of profits expected from spending $5,000 to protect the citrus fruits = $95,000

$96,000 > $95,000

Hence, the citrus grower is better off doing nothing to protect the citrus fruits.

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