Honor Code: You must not discuss these problems with anyone except me. I guarantee that they will be on the final examination, and strongly encourage you to solve them ahead of time (and bring your answers with you to the exam).
You've just gotten a call from Vince Bintner, general manager of the Sonoma Valley winery in which you hold a financial interest. There's been some recent labor unrest in the area, and he feels that there's a 10% chance that a United Farm Workers strike will be called soon.
The grapes aren't quite ready for harvest, and Vince estimates a loss of roughly $250,000 (relative to harvesting at the ideal time) if he begins to harvest the crop early. However, if he waits and a strike is called, he expects a relative loss of about $3,000,000.
Vince tells you that there is just enough time to have a small private survey of UFW workers conducted before he has to make his decision about whether or not to do an early harvest. The survey will estimate whether the workers are currently in a "good" (anti-strike) mood or a "bad" (pro-strike) mood. Of course, moods can change between now and when the strike vote is scheduled. As well, union members don't always truthfully report their mood to outside surveyors. Taking this all into account, Vince tells you that in previous instances when a strike occured, such surveys showed the workers in a bad mood about 70% of the time, while in similar instances when a strike ultimately did not occur, surveys preceding the strike decision showed a good worker mood about 60% of the time.
Vince is waiting to receive a quote on the price of running the survey. If the quote is $75,000, should he have the survey run? What will be his expected cost (relative to harvesting at the ideal time) if he does have the survey run, and uses the resulting information optimally? What is the most he should be willing to pay for the survey?
There are eleven members on the union council. The current council president is Tony Martinez. Of the other ten members, 3 are Tony's relatives, and (independently) each votes the same way Tony does 70% of the time. Of the remaining 7, Vince feels that each will have (independently) about a 60% chance of voting to endorse a strike.
Tony has been making aggressive statements lately, and Vince thinks that, if left to his own devices, Tony has about an 80% chance of voting in support of a strike. However, Vince feels that a nice gift sent to Tony's wife could lower his chance of voting in support of a strike significantly. If Vince doesn't send the gift, what's the probability that a majority (at least 6) of the council members will vote in support of a strike? (Estimate this probability via simulation, using 100,000 observations.)
Vince isn't sure how large a gift would be required to change Tony's vote. As well, there's a 20% chance that Tony is already planning to vote against a strike. (This is the vote that Vince wants.)
Given prior dealings, Vince guesses that a gift worth $20,000 or more will change a pro-strike vote to an anti-strike vote, and less will have no effect. One standard-deviation's-worth of normally-distributed uncertainty in that critical "enough-to-change-a-vote" threshold is $2,500.
Since the outcome of the council vote provides only an initial recommendation to the union members before the full-membership strike vote, Vince figures that having Tony vote against the strike is worth $50,000 more than having him vote in favor of the strike. (In other words, if Vince knew that Tony was going to vote in support of a strike, and could legally pay Tony directly to change his vote, he'd be willing to pay up to $50,000 for the switch.)
How large a gift should Vince send out?
Vince had taken out an insurance policy against harvest difficulties: If the harvest (across his 50 acres) is less than 250 tons, the policy pays off at $3,000/ton of shortfall (e.g., it pays $30,000 for a 240-ton harvest).
With the early harvest, Vince is expecting to bring in 230 tons, with one standard-deviation's-worth of normally-distributed uncertainty being 25 tons. What is his expected payment from the insurance company? (Answer using simulation, with 25,000 observations.)