Page 347, Question 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

Page 347, Answer 4

We have the probability of success is (s) 2/5 with payoff of 55,000 And we have the probability of being unsuccessful of 1 - p(s)= 1 - 2/5 = 3/5 with carry a loss of 1,750.

Hence the expected value of the business venture is follow:

\[ E(b) = \frac{2}{5} \times (55,000) + \frac{3}{5} \times (-1,750) = 20,950 \]

Page 347, Question 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 from selling only one type of beverage. The firm estimates a 1500 profit selling cola if the weather is cold and a 5000 profit of 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% chance 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?

Page 347, Answer 6  

Build a decision tree to assist with the decision

alt text

What should the firm handling concessions do?

E(cola) = .7 x 5000 + .3 x 1500 = 3950
E(coffe) = .7 x 1000 + .3 x 4000 = 1900

Given the expected values above, the firm should be selling cola for the sporting event. Clearly the expected value of cola is more than twice the expected value of the coffee. In fact in sporting events, cola in general is consumed more than coffee as cola is being sold along with hot dog pizza, etc.

Page 355, Question 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 resort has 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.

Page 355, Answer 3

alt text

E(Keep) = .40 x 280000 + .20 x 100000 + .40 x (-40000) = 116000.

We know that if we lease, we will make 100000. Hence we better keep the property.

Page 364, Question 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 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.

Page 364, Answer 3  

alt text

The expected value with survey is as follow:
E( survey, drilling) = (0.6 x 0.85) x ( 6000000 - 1000000 -100000) + # favorable survey of .6 and finding oil .85
(0.6 x 0.15) x (-100000 -1000000)+ # favorable survey of .6 and not finding .15 (0.4 x 0.22) x (6000000 - 100000 - 1000000)+ # unfavorable survey of .4 and finding oil .22 (0.4 x 0.78) x (-1000000 -100000) # unfavorable survey of .4 and not finding oil .78 = 2,488,000

E_survey_drilling = 
 (0.6 * 0.85) * ( 6000000  - 1000000 -100000) +     # favorable survey of .6 and finding oil .85  
(0.6 * 0.15) * (-100000 -1000000)+                  # favorable survey of .6 and not finding .15
(0.4 * 0.22) * (6000000 -100000- 1000000)+          # unfavorable survey of .4 and finding oil .22
(0.4 * 0.78) * (-1000000 -100000)                   # unfavorable survey of .4 and not finding oil .78

E_survey_drilling
## [1] 2488000

The expected value with without survey is as follow: E(no survey, drilling) = (0.45) x (6000000 - 1000000) + (0.55) (-1000000)
= 1,700,000

 E_no_survey_drilling = (0.45) * (6000000 - 1000000) + (0.55) *(-1000000)   
  E_no_survey_drilling
## [1] 1700000

Therefore from the expected values above, the big private company should hire a geologist as the expected value is 2488000 vs 1700000 if they don’t hire geologist before drilling.
Of course, if the big private hired the geologist to perform the survey and they did not drill, they will lose the $100,000 paid to the geologist