The binomial distribution

Some people are addicted to the con. It’s playing the game itself that provides the thrills; the payoff is walking away unscathed. The question is, how long can it keep coming up heads? The answer is until it comes up tails. James William Lewis, the original suspect in the Tylenol case is what you might call an interesting character. He admitted to sending the extortion letters to Johnson and Johnson over the poisoned capsules, but was never convicted to placing the poison himself. He subsequently did a stretch in jail but resumed his life, first as a tax preparer and most recently as a web designer and programmer upon release. Somewhere along the way he was suspected of dismembering a man and raping a woman. However, neither charge stuck.


Now the FBI, some say prompted by publicity attending the anniversary of the crime, others by attentions brought by enterprising minor journalists, is reopening the case. Maybe they have a new tip; maybe modern forensics has provided them with news sources of evidence. explains:

Yesterday, an FBI spokesman in Chicago said that advances in forensic technology, including DNA evidence, had rekindled the investigation. Investigators searching his Cambridge condominium Wednesday were seen carrying out five boxes and a late-model MacIntosh computer.

“As you can imagine if you were at the search scene, there’s a lot of evidence that has to be gone through, a lot of tests that have to take place, and we don’t know if it’s going to be positive,” said the spokesman, Ross Rice.

Investigators had also obtained a warrant to search an unidentified storage facility nearby that Lewis, 62, had rented, according to Rice. A police officer from Arlington Heights, Ill., where three of the slayings occurred, was dispatched to Boston, the Associated Press reported yesterday.

Rice said that police from the Chicago area who investigated the murders still have capsules recovered as evidence. It might be possible to find traces of DNA on them, he said.


Lewis is without a doubt an interesting man; perhaps his problem was that he needed to prove it to the world. In this interview, Roger Nicholson, a Cambridge area journalist who put the spotlight on Lewis again, believed that the so-called Tylenol Man was driven by a need to get back at the world for unspecified offense. He had to keep thumbing his nose in society’s face. Maybe it was all he lived for.

This brings to mind some of the issues arising from the debate between Dick Cheney and former CIA analyst Glen Carle about the danger that Islamic terrorists might sucessfully attack an American city. The relationship is this: it illustrates the power a small probability event that with repetition becomes cumulatively large. If Lewis is found guilty, it will be because he went to the well too often. You might poison 7 people and get away with it. You might even poison 7 people, dismember a man and rape a woman and get away with it. But if you keep at it long enough at some point, some little thing — a fingerprint, some DNA sample, some memory — will bring you down. Even a master criminal has a small probability of getting caught. If he continues long enough, the thousand to one chance may happen.


Small probability threats exist too. And they’re not too bad provided they don’t keep happening. In the Cheney-Carle debate alluded to above, we are told by Carle that al-Qaeda’s chances of launching a catastrophic WMD attack on US soil are small. That may be true, but imagine a criminal organization that survives over decades and keeps buying a large quantity of criminal lottery tickets. What are the odds they eventually makes payday? We won’t remember the hundreds of failures, but we’ll remember the day they get lucky. If Lewis is the the Tylenol killer, he can’t take comfort in all the times he fooled the cops. He’ll remember the one slipup he made.  If you model the problem like that, then it becomes clear that a number of otherwise irrelevant factors become important when trying to prevent a terrorist from eventually hitting the jackpot. The first is to make lottery tickets more expensive, so that the terrorist can’t try too many. The second is to prevent him from learning where he went wrong, so that each attempt has a probability independent of the other. If terrorist — or the real Tylenol Man — can learn where he went wrong, then the probabilities of his individual attempts at success improve over time until finally they become very good indeed.


Has the Tylenol Man taken one thrill pill too many? Time will tell. As for al-Qaeda, when will they stop?


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