Question

DNA Evidence: In a murder trial and expert witness is called to testify about DNA evidence found at the crime scene. They say that the probability of a person randomly matching the DNA fragment found at the crime scene is approximately 1 in a million. In the cross examination the defense attorney asks what the probability of a lab mistake is during the processing of the evidence (contamination, human error, etc), to which the expert witness replies about 5% of the time.

(a) If we are a juror, we are interested in the probability the DNA evidence linking the defendant to the crime scene is false. Lets call this event F. Call the event that the defendant is unluckily enough to share the DNA sequence with the real killer R, and call the event that the lab screwed up and the match is false L. How can we express F in terms of R and L? (b) Using the numbers given by the expert witness find P(F) in this case. (c) Is this beyond your reasonable doubt? How would you explain this to your fellow jurors who aren’t skilled in probability theory?

Answer 1

A. No DNA evidence linkage = F

Rendom matching of DNA Fragment =R = 1/1000,000

Rc= Event of no random matching

False Lab report = L = 5/10

Fc = Event of no false lab report

F = (1000000-1) * 95/100 + 1*5/100= (1000000-R) * (1-L) + (R*L)

B. Prob (F) = Prob (Rc) * Prob (Lc) + Prob (R) * Prob (L)

= 0.999999 * 0.95 + 0.000001*0.05 = 0.9499991

C. To explain it in simple terms the probability of a defendant not matching the DNA at the crime scene is 949999.1 times out of 1 million cases. There are still a chance of 50000.9 or 50001 times out of 1 million times when the defendant can match the DNA test even if he is innocent due to false lab reports or random DNA matching of the defendant to the real killer.