## Xy

Calculate the DNA profile probability for the multilocus genotype using the product rule. The product rule (multiplication rule) is:

Probability of random match = Pm = (Pi)(P2)(P3) . . . (Pn)

Using the data from Table 12.6 will result in the probability:

Pm = (0.034)(0.030)(0.042)(0.128)(0.075)(0.069)(0.082)(0.101)(0.071) (0.014)(0.047)(0.084) (0.123) = 9.395-19

Thus the probability that two people (other than identical twins) have the profile in Table 12.6 is less than one in a hundred trillion.

### Step 5

Calculate confidence limits for each allele. An upper confidence limit should be calculated for each allele frequency in the population. This gives the confidence that the profile is unique, given the population of N unrelated people. The upper 95% confidence limit (95% UCL) has the following formula:

where P is the observed frequency and N is the number of chromosomes studied.

The lower 95% confidence limit (95% LCL) has the following formula:

12.5 Other issues 'Ceiling' principle

In cases where the sample and suspect belong to a subpopulation then a 'ceiling' should be placed on the estimate of the profile. This will be a conservative correction of the estimate. It has been used to compensate for any undetected subpopulation that may exist in the population database.

The 95% UCL should be used or 0.10, whichever is larger. The 95% LCL should be used or 0.05, whichever is smaller.

If a subpopulation (population substructure) database is used, instead of the entire population database, then the ceiling principle would not need to be considered.

The prosecutor's fallacy and defence fallacy

An example of the 'prosecutor's fallacy' is given next. Making a statement like 'There is only a one-in-a-trillion chance that the defendant is innocent' is a statement about guilt or innocence, and is not true. The true statement is 'There is a one-in-a-trillion chance that the forensic sample came from an individual other than the defendant'.

An example of the 'defence fallacy' is as follows. Suppose that a murder occurred in a city with a population of 5 million people. A match was found between the suspect (defendant) and a stain sample from the crime scene. The match probability was calculated to be one in a million. In a city of 5 million, about five people would have a matching profile. Thus the defence argues that the odds are 5 to 1 that the defendant is innocent. This assumes that each of the five people have an equal probability of guilt. This would only be true if the DNA evidence was the only evidence and was used in isolation of any other facts pertaining to the case.

Bayes' theorem

Bayes' theorem may be utilized in select circumstances, but it is not a commonly accepted statistical method for presenting forensic evidence. This method is based on prior probabilities based on certain facts of a specific case. The major argument against using Bayes' theorem is that the prior probabilities may be subjective.

### Likelihood ratio

A commonly accepted way of expressing the likelihood of matching evidence is by calculating the likelihood ratio (LR). Here is an example of a criminal case utilizing the LR method:

P (evidence originated from suspect)

P (evidence originated from an unrelated person in the population)

Probability that the prosecutor is correct

Probability that the defence is correct P (prosecetor's hypothesis)

P (defence hypothesis)

The likelihood that two people are siblings can be calculated like this: P [allele (s) would match if two people were siblings]

P [allele (s) would match if two people were unrelated]