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 are a senior partner in the law firm of Lickem & Winn, and are currently lead counsel in a class-action product liability suit filed against ESU Ltd. Your clients scraped up $200,000 as a retainer, and you agreed to take a 30% contingency fee as the rest of your firm's compensation. (This means that any further expenses are covered completely by your firm, and you will receive 30% of any settlement payment or court award.)
You originally estimated that taking the case to trial would impose costs on your firm (consisting primarily of time that could otherwise be billed to other accounts) of approximately $150,000. Still, you thought there was a 45% chance of winning at trial, and that a victory would generate a $2,000,000 award, so you were expecting (at the time you accepted the case) a net profit of $200,000+45%·30%·$2,000,000-$150,000 = $320,000 from taking on the case and going to trial.
In an attempt to do even better for your firm, you managed to convince your clients to put a $750,000 settlement offer on the table. If accepted, you'd save the cost of going to trial, and make $200,000+30%·$750,000 = $425,000 from your handling of the case. (Your clients, somewhat risk averse, were willing to make the settlement offer rather than risk going to trial.)
Unfortunately, ESU has declined the settlement offer. After holding a mock trial, they've come back with a counter-offer to settle for $400,000. (One of your in-house investigators has been watching the ESU legal staff, and learned of the mock trial.) You infer from ESU's refusal to accept the original settlement offer that their mock trial must have led to a "not liable" finding.
It is well known that mock trials yield the same outcome as a real trial would about 75% of the time. Therefore, you now think there's only a probability of 3/14 (roughly a 21.43% chance) of winning the case at trial. However, your clients have been angered by the refusal, and are now insisting that the case be taken to trial.
You could directly proceed to trial. But your expected gain from doing so is only 3/14·30%·$2,000,000 = $128,571, which is less than the $150,000 cost you would incur. (Of course, taking your retainer into account you still come out ahead.)
You see one possibility: You could hold a mock trial of your own. It would cost you an additional $50,000, and (as always) would have a 75% chance of yielding the same result that a real trial would. If the mock trial returns a finding of "liable," you'll have to go to court - Your clients will insist upon it. However, if the mock trial yields a finding of "not liable," you expect to be able to use that finding to convince your clients to accept the counter-offered $400,000 settlement. (Of course, the outcome of your mock trial would also lead you to update your assessment of your likelihood of winning in court.)
Should you just grit your teeth and go to trial, or should you take on the added expense of holding a mock trial first? What is your firm's expected net profit (include your retainer in your calculations) on this case under each alternative: (1) go to trial immediately vs. (2) hold a mock trial first?
You end up taking the case to trial. This being a civil case, you only need to get 9 (or more) of the 12 jurors to vote "liable" in order to win. As you prepare your closing argument, you feel the trial has gone well for your side.
One of the jurors is a retired physician. As the trial progressed, your jury consultant observed the other jurors frequently looking in her direction, and concludes that her vote will sway the votes of the other 11 jurors. Specifically, if she votes "liable," there's (independently) a 75% chance that each of the others will as well. However, if she votes "not liable," there's only a 50% chance that each of the others will vote "liable."
You intend to pitch your closing argument directly at her: "Imagine that your grandson were brought into the emergency room �" You feel that this will give you an 80% chance of winning a vote of "liable" from her (and will consequently indirectly affect how the other jurors vote).
Using 100,000 simulated observations, (3) estimate the probability that you will win the case by taking this approach. (This problem is analytically tractable; still, please solve it using simulation.) What is the margin of error (at the 95%-confidence level) in your estimate?
You've won the case! Now, the jury is considering the size of the liability award. You estimate that the award will be around $2,000,000, with one standard-deviation's-worth of (normally distributed) uncertainty in your estimate being $300,000.
ESU Ltd has approached you with one last settlement offer: After the jury decides on the award, ESU will instead pay $2,000,000, plus half of the amount by which the jury award exceeds $2,000,000 (for example, if the jury award is $1,800,000, ESU pays $2,000,000; if the jury award is $2,200,000, ESU pays $2,100,000). In return, your clients must agree to not give any post-trial interviews to the press. You have taken this offer to your clients, and they've asked you what the expected total ESU payment (before subtracting off your 30%) will be if they accept the deal.
(4) Estimate this amount using 30,000 simulated observations. (5) Roughly how many observations would be needed to make this estimate with a margin of error (at the 95%-confidence level) of only $250?