## Info

We assume that there are 20 possible suspects, 10 servants and 10 guests, only one of whom is guilty of murdering the owner of an English country estate. This provides the totals in the bottom row. The totals for each column are multiplied by the appropriate probabilities in each cell of Table 3.3 to get the values in this table, which can be used to compute the probability that an innocent suspect is mistakenly identified as guilty. This probability is 0.95/1.31 = 0.73.

We assume that there are 20 possible suspects, 10 servants and 10 guests, only one of whom is guilty of murdering the owner of an English country estate. This provides the totals in the bottom row. The totals for each column are multiplied by the appropriate probabilities in each cell of Table 3.3 to get the values in this table, which can be used to compute the probability that an innocent suspect is mistakenly identified as guilty. This probability is 0.95/1.31 = 0.73.

called posterior probability of guilt is 27%. This is a substantial increase over the 10% prior probability of guilt, but it is far short of the standard expressed by the phrase "beyond a reasonable doubt."6

Suppose we wanted to be 90% sure of blaming the correct person for the crime. How many suspects would we have to exclude based on other evidence so that the posterior probability of guilt following scent identification by the dog was greater than 90%? The best we can do is to reduce the pool of suspects from the original 20 to two, based on other evidence. In this case, after the dog picks one of these suspects, the posterior probability that this suspect is guilty is 88%: not quite reaching the standard for establishing guilt beyond a reasonable doubt .7

In pract ice, other ev idence besides scent ident ificat ion is often qualitative and not easily converted into the prior probabilities used in this example. However, if Schoon's results on the abilities of trained dogs to match human scents are accurate, the method of scent lineups doesn't seem to have much credibility. Even in the ideal and unlikely situation that other evidence reduces the pool of potential suspects to two, there is still about a 12% chance that the dog would pick the wrong suspect from a lineup.

Let's consider a parallel situation in which these kinds of calculations are helpful but actually validate a common forensic method rather than cast ing doubt on it. This is the well-known use of DNA to match suspects to blood or tissue samples found at a crime scene (Gomulkiewicz and Slade 1997). We can set up tables just like Tables 3.3 and 3.4 to show the calculations. With current technology, the likelihood of missing a match between a truly guilty suspect and a sample of that suspect's DNA from a crime scene is very low, certainly less than 0.5% and perhaps in principle equal to zero. This is a false negative or false mismatch, shown in the lower-left cell of Table 3.5. The probability of a false positive depends on four factors: the possibility that an innocent person shares the same DNA profile as the perpetrator of the crime for the regions of DNA that were analyzed, the possibility of laboratory error such as contamination of a sample, the possibility that the innocent person left his or her DNA at the crime scene but did not commit the crime, and the possibility that a DNA sample from the innocent person was planted at the crime scene. If the latter three possibilities can be ruled out, the probability of a false positive ranges from one in 100,000 to one in 1 bill ion, for suspects who are not relatives of the perpetrator of the crime? Because brothers, for example, share 50% of their DNA, the likelihood of a false positive is as high as 0.26 for an innocent suspect who is the brother of the actual criminal. For Table 3.5, I assume that the pool of potential suspects includes only unrelated people, and I use an intermediate value for the probability of a false posit ive identificat ion of one in 10 million.

Applying these values to the murder at the English country estate (and assuming that the mysterious handkerchief has blood on it that does not match that of the victim), Table 3.6 shows that the likelihood of guilt of a suspect whose DNA profile matches that of the blood on the handkerchief is

Table 3.5. Identification of suspects based on blood or tissue samples containing DNA collected at a crime scene.

Status of Suspect

Guilty Innocent

(Suspect = Perpetrator) (Suspect ^ Perpetrator)

Matches Suspect 0.995 0.0000001

Does Not Match

Suspect 0.005 0.9999999

### DNA Collected at Crime Scene

The values in the table are the probabilities that DNA collected at the crime scene (1) matches that of the suspect if the suspect is guilty (the upper-left cell), (2) matches that of the suspect if the suspect is innocent (a false positive result in the upper-right cell), (3) does not match that of the suspect even though the suspect is guilty (a false negative result in the lower-left cell), and (4) does not match that of the suspect if the suspect is innocent (the lower-right cell). These values were derived from a review of the use of DNA evidence in court by Gomulkiewicz and Slade (1997).

0.995/0.9950019, which is greater than 99.99%. Even if the pool of potential suspects was much larger than 20, DNA evidence may be quite persuasive, provided factors like sloppiness in lab techniques or plant ing evidence at the crime scene can be excluded.

However, there may be situations in which even DNA evidence is not as conclusive as might be assumed. Suppose the only evidence available is DNA from a crime scene. The FBI and other law enforcement agencies have databases of DNA profiles for large numbers of individuals who have had various encounters with the legal system. The sizes of these databases are increasing daily. If the authorities have no other evidence, they may scan the database to see if there is a profile that matches that of the DNA from the crime scene. If there are 5 million profiles in the database and we assume that the guilty person is one of those 5 m illion, we would subst itute 5 million for the overall total in the bottom right cell of Table 3.6. In this case, with one guilty person

Table 3.6. An application of DNA identification to a murder in an English country estate.

Status of Suspect

Guilty (Suspect Innocent (Suspect = Perpetrator) ^ Perpetrator) Totals

DNA Collected

Matches Suspect

0 0