Page 347: #4

We have engaged in a business venture. Assume the probability of success is \(P(s)=2/5\); further assume that if we are successful we make $55,000, and if we are unsuccessful we lose $1750. Find the expected value of the business venture.

\(p = \text{probability of success} = .40\)

\[E(BV) = 55000(p) + (-1750)(1 - p)\]

\[E(BV) = 55000*2/5 - 1750 *.60 = 22000 - 1050 = 20950 \]

The expected value of the business venture is $20,950.

Page 347: #6

Consider a firm handling concessions for a sporting event. The firm’s manager needs to know whether to stock up with coffee or cola and is formulating policies for specific weather predictions. A local agreement restricts the firm to selling only one type of beverage. The firm estimates a $1500 profit selling cola if the weather is cold and a $5000 profit selling cola if the weather is warm. The firm also estimates a $4000 profit selling coffee if it is cold and a $1000 profit selling coffee if the weather is warm. The weather forecast says that there is a 30% of a cold front; otherwise, the weather will be warm. Build a decision tree to assist with the decision. What should the firm handling concessions do?

Beverage Decision Tree

Beverage Decision Tree

Based on the decision tree and the calculated expected values, the firm should focus on cola which has a higher expected value than coffee.

Page 355: #3

The financial success of a ski resort in Squaw Valley is dependent on the amount of early snowfall in the fall and winter months. If the snowfall is greater than 40 inches, the resort always has a successful ski season. If the snow is between 30 and 40 inches, the resor thas a moderate season, and if the snowfall is less than 30 inches, the season is poor, and the resort will lose money. The seasonal snow probabilities from the weather service are displayed in the following table with the expected revenue for the previous 10 seasons. A hotel chain has offered to lease the resort during the winter for $100,000. You must decide whether to operate yourself or lease the resort. Build a decision tree to assist in the decision.

Type Snow >= 40 in. (\(p_1=0.40\)) Snow 30-40 in (\(p_2=0.20\)) Snow <= 30 in (\(p_3=0.40\))
Operate $280,000 $100,000 -$40,000
Lease $100,000 $100,000 $100,000
Operate/Lease Decision Tree

Operate/Lease Decision Tree

Using the decision tree to calculate the expected values, E(Operate) = $116,000 which is greater than E(Lease) = $100,00. Therefore, it would make sense to operate the resort instead of leasing during the winter.

Page 364: #3

A big private oil company must decide whether to drill in the Gulf of Mexico. It costs $1 million to drill, and if oil is found its value is estimated at $6 million. At present, the oil company believes that there is a 45% chance that oil is present. Before drilling begins, the big private oil company can hire a geologist for $100,000 to obtain samples and test for oil. There is only about a 60% chance that the geologist will issue a favorable report. Given that the geologist does issue a favorable report, there is an 85% chance that there is oil. Given an unfavorable report, there is a 22% chance that there is oil. Determine what the big private oil company should do.

Drill Decision Tree

Drill Decision Tree

\[E(Drill-No Survey) = 0.45 * (6000000 - 1000000) + 0.55 * -1000000 = $1,700,000\]

\[\begin{eqnarray} E(Hire Geologist) = (0.6)*(0.85)*(6000000 - 1000000 - 100000) + \\ (0.6)*(0.15)*(- 1000000 - 100000) + \\ (0.4)*(0.22)*(6000000 - 1000000 - 100000) + \\ (0.4)*(0.78)*(- 1000000 - 100000) = $2,488,000 \end{eqnarray}\]

The private oil company should definitely hire the geologist with the expected value \(E(Hire Geologist) = \$2,488,000\)

Page 373: #1

Given the following payoff matrix, show all work to answer parts (a) and (b).

  1. Which alternative do we choose if our criterion is to maximize the expected value?
AltA <- (0.35) * (1100) + (0.3) * (900)  + (0.25) * (400)  + (0.1) * (300)
AltB <- (0.35) * (850)  + (0.3) * (1500) + (0.25) * (1000) + (0.1) * (500)
AltC <- (0.35) * (700)  + (0.3) * (1200) + (0.25) * (500)  + (0.1) * (900)

alternative <- c(AltA, AltB, AltC)

max(alternative)
## [1] 1047.5

Alternative B should be chosen to maximize the expected value with \(E(B) = 1047.5\)

  1. Find the opportunity loss (regret) table and compute the expected opportunity loss (regret) for each alternative. What decision do you make if your criterion is to minimize expected regret?

Create the regret (or cost) matrix by subtracting each entry from the largest entry in each column. Then multiply each column by it’s probability.

AltA <- c(1100-1100, 1500-900, 1000-400,900-300)
AltB <- c(1100 - 850, 1500 - 1500, 1000-1000, 900-500)
AltC <- c(1100-700, 1500-1200, 1000-500, 900-900)

(res <- rbind(AltA, AltB, AltC) %*% c(0.35, 0.3, 0.25, 0.1))
##       [,1]
## AltA 390.0
## AltB 127.5
## AltC 355.0

We see that alternative B has the lowest cost with \(E(B) = 127.5\)