The Math of Confused Eyewitnesses

Feb. 1 -- "Yes, he's definitely the one I saw that night. I'll never forget that sneer."

Confident, but mistaken eyewitness reports during criminal trials can send an innocent man or woman to prison. And new experiments (as well as common sense) indicate that such faulty identifications of suspects are not uncommon.

Before I get to developments on this topic, consider a coin puzzle whose solution is relevant to the issue.

Picking the Biased Penny

Assume that you have three suspect pennies lined up before you. You're told that one of these pennies, the culprit, lands heads 75 percent of the time, and that the other two, the innocent suspects, are fair coins. You know nothing else about the pennies, but you did previously observe that one of the coins was flipped three times and landed heads all three times.

Having witnessed this and realizing that the biased penny is much more likely to behave in this way, you identify this coin as the culprit.

How likely are you to be right?

If you were randomly to pick a penny from the lineup, the probability that it would bethe culprit would be 1/3 or about 33 percent. But given that you have this (less than conclusive) information about one of the pennies, what is the probability that it is the culprit?

The answer to the problem, obtained using what is known as Bayes Theorem(see the sidebar below if you must know how), is 63 percent. We revise our probability estimate of that penny's being the culprit upward from 33 percent to 63 percent because it's been flipped three times and has landed heads all three times.

The calculations are formally analogous to what we do when we change our estimate of the probability of a suspect's guilt after the testimony of an eyewitness. Identifying a biased coin on the basis of the evidence of three consecutive heads is mathematically the same as identifying a human culprit on the basis of an eyewitness' memory.

Picking the Wrong Guy

There are, of course, many complicating issues in the case of eyewitnesses and suspect lineups. An article by Atul Gawande in the Jan. 8 edition of The New Yorker details the work of Gary Wells, a psychologist at Iowa State University, and others who have noted the alarming error rate amongeyewitnesses to crimes. They have discovered a number of factors that significantly influence the likelihood that witnesses will correctly pick the culprit out of a lineup.