A Real World Example

Forensic thinking is best explained with a real world example. As unpleasant as they are, rape cases often involve straightforward forensic thinking. I apologize in advance for anyone who might be upset by my blunt and clinical discussion of an extremely painful crime. A male assailant in a typical sexual assault leaves semen either on or inside the body of the victim. The assailant's DNA can be extracted from the semen and a DNA profile determined. If the DNA profile matches a suspect, the analyst must now determine what this match really means.

The forensic analyst asks two questions:

How likely is the DNA profile we found if we assume the suspect committed the assault?

vs.

How likely is the DNA profile we found if we assume a random person committed the assault?

The first question is simple. If the person whose DNA profile matches the profile recovered from the victim committed the rape, we'd expect to find his DNA profile every time. Therefore, the probability of his profile being found at a rape he committed would be one.

To determine how likely the DNA profile would be if someone else committed the crime, we need to know how often we'd expect to find a person unrelated to the suspect with a DNA profile matching the crime scene profile.

Population studies allow us to estimate the rarity of a DNA profile. In this case the crime scene profile is quite rare, the probability of encountering it is 0.000000000001. How small of a probability is that? Imagine the odds of rolling a "6" fifteen times in a row on a standard six-sided die. Imagine flipping a coin and have it turn up heads 147 times in a row. Get the general idea?

With a solid estimate of how rare the crime scene profile is, we can now evaluate our two scenarios.

The probability of encountering the DNA profile if the suspect committed the rape: 1

The probability of encountering the DNA profile if some random person committed the rape: 0.000000000001

Computing a ratio of the probabilities: 1/0.000000000001 = 1,000,000,000,000 or roughly one trillion to one.

In this case, the DNA evidence we found is one trillion times more likely if we assume that the suspect committed the rape than if we assume that some unrelated person did the deed.

Notice how the alternate hypothesis uses the word "unrelated" person. What if the suspect has a sibling or other blood relative that may have also committed the crime? Then our calculation changes significantly, because the two scenarios become:

The probability of encountering the DNA profile if the suspect committed the rape

vs

The probability of encountering the DNA profile if a sibling of the suspect committed the rape

Thinking clearly about our assumptions is absolutely critical. For example, imagine if the suspect and victim were romantically involved. Finding a DNA profile matching the suspect would not be a surprise. The crime scene profile would not provide any useful information under such a scenario.