# The Math of Confused Eyewitnesses

This is not a trivial problem, since it's estimated that almost 80,000 people annually become criminal defendants after being picked out of a lineup by eyewitnesses.

Calculating the Accuracy of a Witness

Using Bayes' Theorem, Wells points out that the base rate likelihood — the initial probability that a suspect is the culprit — greatly affects the subsequent likelihood — the probability that he is the culprit given that he has been picked out of a lineup. If the base rate likelihood is small, eyewitness' identifications need to be almost certain to ensure the subsequent probability is reasonably high. In the penny problem discussed above, the base rate likelihood of picking the culprit is 33 percent, the subsequent likelihood 63 percent.

Another useful notion that Wells explores is the information gain from eyewitness identification. This is the difference between the subsequent likelihood that he is the culprit given he's been identified as such and the base rate likelihood that he is.

Again considering the biased penny to be a culprit of sorts, we conclude the information gain from seeing a penny flipped three times and landing heads all three times is the difference between the probability the penny is the culprit given that it's landed heads three times (63 percent) and the initial probability the penny is the culprit (33 percent). The information gain is thus 30 percent.

There are, of course, many other nuances and variations but the bottom line is that the eyewitness testimony on which people are convicted is sometimes not worth three cents.

Professor of mathematics at Temple University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on ABCNEWS.com appears on the first day of every month.

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