Hollywood Thrills
10.1 THE PROBLEM
Shortly after his death in Biloxi, Mississippi, in a tragic bungee-jumping accident, the following preliminary movie workup was found in the papers of famed Hollywood director Irving Nutso. Nutso, who as a teenager flunked out of Caltech because he spent all his time watching movies at virtually every theater in the Pasadena, Cucamonga, and Azusa areas around the institute instead of attending classes, was at the time of his death planning his comeback film. With the working title of Revenge of the Ducks, it would have been his return to the silver screen after being released from federal prison, having served ten years for alleged involvement with foreign gangsters (who were never charged) in a break-in at the Beverley Hills branch of the International Bank of the Stars.
The millions stolen during that robbery were never recovered, and authorities suspected that Nutso had secretly hidden a lot of cash without telling his gangster friends where. But nothing was ever proved. Still, it might explain where Nutso got funding for Revenge (as well as why a 500-foot bungee cord was “accidently” substituted for the correct one when Nutso jumped from a 400-foot-high bridge). While it remains unclear if the film will ever be made, Nutso’s preliminary workup does present us with an interesting math problem.
A terrible, never-before-seen bacterial infection has suddenly appeared, worldwide. It seems to have initially appeared in unsanitary theater popcorn butter dispensers, but now it’s everywhere. No one is safe, certainly not in airport pay-phone booths or even on Capitol Hill. (Note to Special Effects: Can we show some shots of Congress-people running screaming from their offices using stock footage, or will we have to actually hire some senators?) Once people begin to exhibit symptoms, two-thirds die a terrible, drawn-out death, the specifics of which are so horrible as to be almost beyond imagination (but not completely, as close-up details can be graphically depicted in a pre-release movie trailer). Then, just when all hope seems lost, not one but two new experimental drugs are announced, each by a different medical team. (All the doctors on each team are beautiful and/or handsome, with two exceptions, as well as all being under the age of 25, even though every one of them has an MD and a PhD—the exceptions are that each team will include a goofy but charmingly funny, wise-cracking computer whiz).
Unfortunately each team’s preliminary, very uncertain results are based only on animal studies, specifically from lab ducks, as the duck is the only nonhuman animal in which the popcorn butter–loving bacteria thrive. So each drug is separately given to a different group of infected humans. It’s impossible to give both drugs to the same person as, together, the drugs interact to form a compound that dissolves bone tissue, a scenario we can gruesomely illustrate in a movie trailer by showing vast flocks of lab ducks collapsing into hysterically quacking blobs as their skeletons turn to mush. This should be quite impressive in 3-D.
The duck data must therefore be supplemented by risky human tests, and so a small number of brave, infected people step forward to serve as volunteers for two fast-result, emergency tests. When drug no. 1 is given to a test group of five infected people, everybody recovers. When drug no. 2 is given to a test group of 12 infected people, however, three die. The president of the United States is then faced with the monster decision of which drug to launch into massive production. Time and money are available only for one or the other. What to do?
The hero of the movie will be portrayed as a brilliant (we’ll give him three PhDs!) academic mathematician on a sabbatical leave from DUCKSTEW (the Distinguished University for Combining Knowledge with Scientific Technology for the Entire World), who was recently appointed to be special math assistant to the president; it is his enormous task to advise the president on which drug to choose. (I think this role should be played by a well-known teenage heart-throb, who will help attract a younger audience to the film.)
Nutso’s notes end here, with just a final scribbled note at the bottom in pencil, saying, “More later, as soon as I get back from my big jump!” We know how that ended. How sad.
Anyway, here’s the puzzle for you: if you were on sabbatical from DUCKSTEW as the president’s math adviser, which of the two drugs would you support, the one that had no deaths in its trials or the one that had three deaths? The fate of the world—and the academic reputation of DUCKSTEW—hang in the balance! Only a Hollywood genius like Irv Nutso would have dared to make a gutsy movie like this!
10.2 THEORETICAL ANALYSIS
Most people would be emotionally inclined to go with drug no. 1, which had zero deaths in its test group. That’s a perfect record! But, as a DUCKSTEW math prof, you know better: it’s actually drug no. 2, with three deaths in its test group, that should be chosen. Here’s why. Let’s suppose that each of the drugs is actually worthless. That would mean the outcome for each of the test groups was due only to random chance. What we’ll do, then, is first calculate the probability that, for each test group, what actually occurred was due only to chance. Then we’ll ask ourselves the following question: Which do we believe, “random chance is behind what happened” or “the hypothesis that the drug was worthless has such a small probability—the probability we just computed—that it is more reasonable to reject that hypothesis in favor of believing that the drug did have value”? Our choice, then, will be the drug whose trials had the smaller probability of occurring only by chance. That is, we’ll choose the drug that has the greater likelihood of supporting the rejection of the “worthless” hypothesis.
Let q be the probability an infected person dies after receiving a worthless drug. That means q = 2/3. If out of five people all survive (that is, there are zero deaths), then we have an event with probability
If out of 12 people at most three die, then we have an event with probability
It is a close call, but even with three deaths during its trials, drug no. 2 offers a slightly better chance of being the more effective drug than does drug no. 1. It’s a nonintuitive conclusion, yes, but there it is, a nice illustration of the power of mathematical analysis.
See you at the movies, but probably not at Revenge of the Ducks. To be on the safe side, however, just remember to skip putting any butter on your popcorn! And say a prayer for Irv.