A Field Guide to Lies: Critical Thinking in the Information Age by Daniel J. Levitin Page A
Authors: Daniel J. Levitin
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more accurate estimate. For example, if your father and mother both had lung cancer, you want to calculate the probability of you contracting lung cancer by looking at other people in this select group, people whose parents had lung cancer. If your parents didn’t have lung cancer, you want to look at the relevant subgroup of people who lack a family history of it (and you’ll likely come up with a different figure). If you want to know the probability that your waiter will bring you ketchup, you might look at only the tables of those patrons who ordered hamburgers or fries, not those who ordered tuna tartare or apple pie.
Ignoring the dependence of events (assuming independence) can have serious consequences in the legal world. One wasthe case of Sally Clark, a woman from Essex, U.K., who stood trial for murdering her second child. Her first child had died in infancy, and his death had been attributed to SIDS (sudden infant death syndrome, or crib death). The prosecutors argued that the odds of having two children die of SIDS were so low that she must have murdered the second child. The prosecution’s witness, a pediatrician, cited a study that said SIDS occurred in 1 out of 8,543 infant deaths. (Dr. Meadow’s expertise in pediatrics does not make him an expert statistician or epidemiologist—this sort of confusion is the basis for many faulty judgments and is discussed in Part 3 of this book; an expert in one domain is not automatically an expert in another, seemingly related, domain.)
Digging deeper, we might question the figure of 8,543 deaths. How do they know that? SIDS is a diagnosis of exclusion—that is, there is no test that medical personnel can perform to conclude a death was by SIDS. Rather, if doctors are not able to find the cause, and they’ve ruled out everything else, they label it SIDS. Not being able to find something is not proof that it didn’t occur, so it is plausible that some of the deaths attributed to SIDS were actually the result of less mysterious causes, such as poisoning, suffocation, heart defect, etc.
For the sake of argument, however, let’s assume that SIDS is the cause of 1 out of 8,543 infant deaths as the expert witness testified. He further testified that the odds of two SIDS deaths occurring in the same family were 1⁄8543 x 1⁄8543, or 1 in 73 million. (“Coincidence? I think not !” the prosecutor might have shouted during his summation.) This calculation—this application of the multiplication rule—assumes the deaths are independent, but they might not be. Whatever caused Mrs. Clark’s first child to die suddenly might be present for both children by virtue of them being in the same household: Two environmental factors associated with SIDS are secondhand smoke and putting a baby to sleep on its stomach. Or perhaps the first child suffered from a congenital defect of some sort; this would have a relatively high probability of appearing in the second child’s genome (siblings share 50 percent of their DNA). By this way of thinking, there was a 50 percent chance that the second child would die due to a factor such as this, and so now Mrs. Clark looks a lot less like a child murderer. Eventually, her husband found evidence in the hospital archives that the second child’s death had a microbiological cause. Mrs. Clark was acquitted, but only after serving three years in prison for a crime she didn’t commit.
There’s a special notation for conditional probabilities. The probability of a waiter bringing you ketchup, given that you just ordered a hamburger, is written:
P(ketchup | hamburger)
where the vertical bar | is read as given . Note that this notation leaves out a lot of the words from the English-language description, so that the mathematical expression is succinct.
The probability of a waiter bringing you ketchup, given that you just ordered a hamburger and you asked for the ketchup, is noted:
P(ketchup | hamburger ∧ asked)
where the ∧ is read as and
Ignoring the dependence of events (assuming independence) can have serious consequences in the legal world. One wasthe case of Sally Clark, a woman from Essex, U.K., who stood trial for murdering her second child. Her first child had died in infancy, and his death had been attributed to SIDS (sudden infant death syndrome, or crib death). The prosecutors argued that the odds of having two children die of SIDS were so low that she must have murdered the second child. The prosecution’s witness, a pediatrician, cited a study that said SIDS occurred in 1 out of 8,543 infant deaths. (Dr. Meadow’s expertise in pediatrics does not make him an expert statistician or epidemiologist—this sort of confusion is the basis for many faulty judgments and is discussed in Part 3 of this book; an expert in one domain is not automatically an expert in another, seemingly related, domain.)
Digging deeper, we might question the figure of 8,543 deaths. How do they know that? SIDS is a diagnosis of exclusion—that is, there is no test that medical personnel can perform to conclude a death was by SIDS. Rather, if doctors are not able to find the cause, and they’ve ruled out everything else, they label it SIDS. Not being able to find something is not proof that it didn’t occur, so it is plausible that some of the deaths attributed to SIDS were actually the result of less mysterious causes, such as poisoning, suffocation, heart defect, etc.
For the sake of argument, however, let’s assume that SIDS is the cause of 1 out of 8,543 infant deaths as the expert witness testified. He further testified that the odds of two SIDS deaths occurring in the same family were 1⁄8543 x 1⁄8543, or 1 in 73 million. (“Coincidence? I think not !” the prosecutor might have shouted during his summation.) This calculation—this application of the multiplication rule—assumes the deaths are independent, but they might not be. Whatever caused Mrs. Clark’s first child to die suddenly might be present for both children by virtue of them being in the same household: Two environmental factors associated with SIDS are secondhand smoke and putting a baby to sleep on its stomach. Or perhaps the first child suffered from a congenital defect of some sort; this would have a relatively high probability of appearing in the second child’s genome (siblings share 50 percent of their DNA). By this way of thinking, there was a 50 percent chance that the second child would die due to a factor such as this, and so now Mrs. Clark looks a lot less like a child murderer. Eventually, her husband found evidence in the hospital archives that the second child’s death had a microbiological cause. Mrs. Clark was acquitted, but only after serving three years in prison for a crime she didn’t commit.
There’s a special notation for conditional probabilities. The probability of a waiter bringing you ketchup, given that you just ordered a hamburger, is written:
P(ketchup | hamburger)
where the vertical bar | is read as given . Note that this notation leaves out a lot of the words from the English-language description, so that the mathematical expression is succinct.
The probability of a waiter bringing you ketchup, given that you just ordered a hamburger and you asked for the ketchup, is noted:
P(ketchup | hamburger ∧ asked)
where the ∧ is read as and
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