She didn't think a jury would be willing to fault the company for taking the 98-percent side of the bet, especially given the stakes. Even if she argued that the company should have re-tested Jake to give him the benefit of the 2-percent doubt, she knew the company's lawyers could argue convincingly against her. They would no doubt claim that drug users have a variety of techniques to avoid detection, and, now on the alert, Jake could use them to evade follow-up tests.
Nevertheless, she met with a mathematics professor at a local college to review her case. The professor immediately found a fundamental flaw in the company's logic – a flaw that tipped the scales dramatically in Jake's favor.
The professor said that the 98-percent reliability of the positive test result didn't mean that Jake was 98-percent likely to be a heroin user. Rather, it provided incremental evidence to support the claim that Jake used heroin, but what the company needed to reasonably terminate Jake was overwhelming total evidence. He said that in order to determine the total evidence against Jake, the company must take into account prior knowledge about Jake's likelihood of using heroin.
Shelia retrieved the company's yellow fact sheet from her briefcase and asked the professor if it would help.
"Yes," he said, looking it over, "I would say this is exactly what you want." He chuckled, then added, "It says in the opening sentence that 'only 2 out of 1,000 construction workers use heroin.' If that's right, the company made a rather significant error in its interpretation of the statistical evidence."
He said that the company's own statistics show that out of 1,000 construction workers, a whopping 998 of them will not be heroin users. Nevertheless, if you test those 998 clean workers for heroin, 2 percent of them – 19.96 on average – will test positive anyway. Of the 2 actual heroin users, 1.96 (on average) will test positive, too. In total, then, 21.92 people will test positive, but only 1.96 will truly be heroin users. Thus, he concluded, if you divide 1.96 by 21.92, you'll arrive at the proportion of workers who tested positive that actually are heroin users. That proportion turns out to be 8.9 percent.
Sheila couldn't believe it: "Are you saying that Jake's chance of being innocent wasn't 2 percent but in reality greater than 90 percent?!"
"That's exactly what I'm saying."
In the professor's sworn written statement, he put it more formally:
What we are interested in is the probability that a worker actually uses heroin, given than he has tested positive for heroin use. If we let H denote the event that a worker uses heroin, and T denote the event that a worker tests positive for heroin, then we wish to find the probability of H given T, which is written as follows:With the professor's written testimony, Shelia was able to convince the company of its mistake (and the difficulty of defending the mistake in court). The settlement was reached soon thereafter.P(H|T)What we already know is the following: First, the probability of a construction worker using heroin is 0.2 percent. Second, the probability of the test being correct is 98 percent. Thus the probability of testing positive for heroin, given that heroin is in fact used, is 98 percent. Similarly, we can compute the odds of not testing positive (which we will denote ~T) given that heroin is not used (~H). From these, we can compute the probabilities of all the other possibilities. More formally:P(H) = 0.002Now, we can use Reverend Thomas Bayes's famous rule for posterior probabilities to compute our desired value:
P(~H) = 0.998
P(T|H) = P(~T|~H) = 0.98
P(~T|H) = P(T|~H) = 0.02P(H|T) = P(T|H) P(H) / [ P(T|H) P(H) + P(T|~H) P(~H) ]Substituting our probabilities from above yields the answer we seek:P(H|T) = 0.98 * 0.002 / [ 0.98 * 0.002 + 0.02 * 0.998 ] = 0.089Thus, even though Jake tested positive, it was only 8.9 percent likely that he was a heroin user. The overwhelming preponderance of the evidence – over 91 percent – supported the opposite conclusion, that Jake did not use heroin.
Thus ends our two-part series on Jake, part of the Mathematics Can Be Fun And Even Dramatic program. The goal of the program is to replace television's popular criminal justice–courtroom drama with a new kind of show: the criminal justice–statistical-courtroom drama. Please join us in our noble quest! A donation in the amount of 1/sqrt(–π) USD would be most appreciated.
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